Creating and Graphing Two Variable Equations - Algebra 2

1. A concert venue sells tickets. Revenue is $R=25q$ and cost is $C=10q+800$, where $q$ is the number of tickets sold. Create an equation for profit $P$ in terms of $q$, and choose appropriate axes labels and a reasonable scale to graph $P$ for $0\le q\le 200$.
   Answer: Equation: $P=25q-(10q+800)=15q-800$. Graph setup: x-axis Tickets sold (q), y-axis Profit (dollars); x-scale 0 to 200 by 20, y-scale -1000 to 2500 by 500. This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For profit calculations, Profit = Revenue - Cost, so P = R - C = 25q - (10q + 800) = 25q - 10q - 800 = 15q - 800. The equation structure mirrors the context structure! Profit is always revenue minus cost. Given R = 25q (revenue of $25 per ticket) and C = 10q + 800 (variable cost of $10 per ticket plus fixed cost of $800), we calculate P = R - C = 25q - (10q + 800) = 25q - 10q - 800 = 15q - 800. Note the parentheses are crucial when subtracting! For q = 0, P = -800 (loss); for q = 200, P = 15(200) - 800 = 3000 - 800 = 2200. The y-scale needs to show from -1000 to 2500. Choice A correctly creates the equation P = 15q - 800 with proper graph setup: x-axis 'Tickets sold (q)', y-axis 'Profit (dollars)', and scale accommodating both negative values (losses) and positive values (profits). Choice D incorrectly adds 800 instead of subtracting it, missing the fact that the $800 is a cost that reduces profit. Equation creation framework: (1) Define your variables clearly—'Let q = number of tickets sold, P = profit in dollars'—being specific prevents confusion, (2) Identify the mathematical structure: Profit = Revenue - Cost, where subtraction of the entire cost expression requires parentheses, (3) Build the equation: P = 25q - (10q + 800) = 25q - 10q - 800 = 15q - 800, (4) Verify: at break-even, 15q - 800 = 0, so q = 800/15 ≈ 53.3 tickets, which makes business sense! Graph scale decision process: (1) Find your data range—for q from 0 to 200: P(0) = -800, P(100) = 700, P(200) = 2200, (2) Y-values range from -800 to 2200, so use -1000 to 2500 for clarity, (3) Range of 3500, divide by 7 to get 500, (4) Mark intervals: -1000, -500, 0, 500, 1000, ..., 2500. This shows losses and profits clearly!
2. A theater sells tickets for $18 each. The theater also charges a flat online processing fee of $4 per order (no matter how many tickets). Write an equation for total cost $T$ (dollars) in terms of number of tickets $n$ (tickets), and choose appropriate axes labels and scales to graph for $0\le n\le 12$. Which choice is correct?​
   Answer: Equation: $T=18n+4$; Graph: x-axis Tickets (tickets), y-axis Total cost ($); x-scale 0 to 12 by 1, y-scale 0 to 220 by 20 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost (C), independent quantities are hours (h) and items (n), and the relationship is additive with rates: C = 40h + 15n. The equation structure mirrors the context structure! The context describes $18 per ticket plus a flat $4 fee, so total cost T = 18n + 4, with n as independent (tickets) and T dependent; for graphing 0 to 12 tickets, x-axis tickets (tickets), y-axis total cost ($), x-scale 0-12 by 1 to show each unit, y-scale 0-220 by 20 since max is 18*12 + 4 = 220, providing clear increments. Choice A correctly creates the equation T=18n+4 with proper variables and coefficients, and sets up the graph with detailed labels including units and scales that fit the data range effectively. A distractor like choice B swaps the coefficients, maybe confusing the per-ticket cost with the fee, but gently correct by matching 'each' to the rate multiplier and 'flat' to the added constant. Equation creation framework: (1) Define your variables clearly—'Let n = number of tickets (tickets), T = total cost (dollars)'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Graph scale decision process: (1) Find your data range—what are the minimum and maximum values you need to show for each axis? (2) Divide that range by 5-10 to get interval size, (3) Round to a 'nice' number: use 1, 2, 5, 10, 20, 50, 100, etc. (not 7 or 13!), (4) Mark intervals starting at 0 (or other logical point). Example: data from 0 to 80 → range is 80, divided by 8 gives 10, so mark every 10: 0, 10, 20, ..., 80. Clean, readable, shows all data!
3. A delivery company's daily cost depends on miles driven and the number of deliveries. It charges a fixed daily fee of $40 plus $0.60 per mile $m$ and $2.50 per delivery $d$. Let $C$ be total cost (dollars). Create an equation representing the relationship, and choose appropriate variable labels (with units) for the graph of $C$ versus $m$ when $d$ is fixed.
   Answer: Equation: $C=40+0.60m+2.50d$; Graph with x-axis: Miles $m$ (miles), y-axis: Cost $C$ (dollars) for a fixed $d$. This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost ($C$), independent quantities are hours ($h$) and items ($n$), and the relationship is additive with rates: $C = 40h + 15n$. The equation structure mirrors the context structure! The cost is fixed $40 plus $0.60 per mile $m$ plus $2.50 per delivery $d$, so $C = 40 + 0.60m + 2.50d$; for fixed $d$, graph $C$ versus $m$ with x-axis Miles $m$ (miles), y-axis Cost $C$ (dollars). Choice A correctly creates the equation $C = 40 + 0.60m + 2.50d$ and sets up the graph with proper labels for fixed $d$. Choice D swaps the coefficients for $m$ and $d$, which would mismatch the rates; double-check that each 'per' rate multiplies the correct variable. Equation creation framework: (1) Define your variables clearly—'Let x = [exactly what it represents] in [units]'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Proper graph setup requires three things: (1) axis labels that include both variable name AND units ('Time (hours)' not just 't'), (2) scale with intervals fitting your data range (if values go 0-100, mark every 10 or 20, not every 1 or every 100), (3) clear indication of what each axis represents. A well-labeled graph communicates information immediately—anyone should be able to read your graph without additional explanation!
4. The table shown displays the relationship between the side length $s$ of a square and its area $A$. Based on this data, which equation correctly models the relationship, and what would be the most appropriate viewing window for graphing this function?
   Answer: $A = s^2$; window: $0 \leq s \leq 8$, $0 \leq A \leq 65$ For a square, area equals side length squared: A = s². The table confirms this with values like s = 3 giving A = 9, s = 5 giving A = 25, etc. The viewing window 0 ≤ s ≤ 8, 0 ≤ A ≤ 65 accommodates the data range appropriately. Choice B suggests a linear relationship. Choice C adds an unnecessary constant. Choice D modifies the basic square area formula incorrectly.
5. A rectangle has length $(x+4)$ meters and width $(x-1)$ meters. 1) Write an equation for the area $A$ (square meters) in terms of $x$. 2) Then choose appropriate axes labels and a reasonable scale to graph $A$ versus $x$ for $1\le x\le 8$. Which choice is correct?​
   Answer: Equation: $A=x^2+3x-4$; Graph: x-axis $x$ (meters), y-axis Area (m$^2$); x-scale 1 to 8 by 1, y-scale 0 to 120 by 10 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost (C), independent quantities are hours (h) and items (n), and the relationship is additive with rates: C = 40h + 15n. The equation structure mirrors the context structure! The area A = (x+4)(x-1) = x² + 3x - 4, quadratic from multiplying dimensions; graph with x-axis x (meters) independent, y-axis area (m²) dependent, x-scale 1-8 by 1, y-scale 0-120 by 10 to cover max A at x=8 of 84 with room. Choice A correctly creates the equation A=x²+3x-4 by expanding properly and sets up the graph with precise labels and scales. Distractors like choice B subtract instead of add terms, possibly from sign errors in expansion, but verify by FOIL: (x+4)(x-1) = x² -x +4x -4 = x² +3x -4. Equation creation framework: (1) Define your variables clearly—'Let x = variable (meters), A = area (m²)'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Graph scale decision process: (1) Find your data range—what are the minimum and maximum values you need to show for each axis? (2) Divide that range by 5-10 to get interval size, (3) Round to a 'nice' number: use 1, 2, 5, 10, 20, 50, 100, etc. (not 7 or 13!), (4) Mark intervals starting at 0 (or other logical point). Example: data from 0 to 80 → range is 80, divided by 8 gives 10, so mark every 10: 0, 10, 20, ..., 80. Clean, readable, shows all data!
6. A water tank is being drained at a rate that depends on the height of water remaining. The relationship between time $t$ (in minutes) and height $h$ (in feet) is inverse: as time increases, height decreases such that their product remains constant at 240. Which equation and graph characteristics correctly describe this situation?
   Answer: $h = \frac{240}{t}$ for $t > 0$; hyperbola in first quadrant, decreasing function When you encounter problems describing inverse relationships, look for the key phrase that their "product remains constant." This signals an inverse variation where one variable equals a constant divided by the other. Since the problem states that time and height have a product that remains constant at 240, we can write $ht = 240$. Solving for height gives us $h = \frac{240}{t}$, which is the standard form of inverse variation. Since we're dealing with a real-world scenario involving time and water height, both variables must be positive, so $t > 0$. This creates a hyperbola in the first quadrant only. As time increases, height decreases, making this a decreasing function. Choice A represents linear decay ($h = 240 - t$), not inverse variation. While it does show height decreasing over time, the relationship isn't inverse since the product $ht$ wouldn't be constant. Choice C correctly identifies the relationship as $ht = 240$ but incorrectly suggests the graph exists in all four quadrants. In this context, negative time or negative height don't make physical sense. Choice D has the right equation and domain but incorrectly describes the function as increasing. When $h = \frac{240}{t}$, as $t$ increases, $h$ must decrease since they're inversely related. The answer is B: $h = \frac{240}{t}$ for $t > 0$, creating a decreasing hyperbola in the first quadrant. Study tip: When you see "product remains constant" or "inversely related," immediately think $xy = k$ or $y = \frac{k}{x}$. Always consider the practical domain restrictions based on the real-world context.
7. A bike rental shop charges a flat fee of $12 plus $4 per hour. Write an equation for the total cost $C$ (dollars) in terms of hours rented $h$ (hours), and choose appropriate axes labels and a reasonable scale to graph the relationship for $0\le h\le 8$.
   Answer: Equation: $C=12+4h$; x-axis: Hours (hours); y-axis: Cost ($); scale: x 0–8 by 1, y 0–50 by 5 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $12 flat fee plus $4 per hour,' the dependent quantity is cost (C), independent quantity is hours (h), and the relationship is additive with rates: C = 12 + 4h. The equation structure mirrors the context structure! The flat fee of $12 is the starting cost (y-intercept), and $4 per hour means multiply hours by 4, giving C = 12 + 4h. Choice B correctly creates the equation C = 12 + 4h with proper axis labels including units (Hours (hours) and Cost ($)) and a reasonable scale showing 0-8 hours horizontally and 0-50 dollars vertically. Choice A incorrectly reverses the equation to h = 4C + 12, making hours depend on cost instead of cost depending on hours—this doesn't match the real-world situation! Equation creation framework: (1) Define your variables clearly—'Let h = hours rented, C = total cost in dollars'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'flat fee' means constant term, 'per' means multiply (rate), 'plus' means add, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: for h = 2 hours, C = 12 + 4(2) = $20, which makes sense!
8. A charity sells tickets for a fundraiser. Adult tickets cost $12 each and student tickets cost $8 each. Let $R$ be total revenue (dollars), $a$ be the number of adult tickets, and $s$ be the number of student tickets. Write an equation representing this relationship, and choose correct axes labels for a graph of $R$ versus $a$ when $s$ is fixed at $20$.
   Answer: Equation: $R=12a+8s$; To graph with $s=20$, use x-axis: Adult tickets $a$ (tickets), y-axis: Revenue $R$ (dollars). This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost ($C$), independent quantities are hours ($h$) and items ($n$), and the relationship is additive with rates: $C = 40h + 15n$. The equation structure mirrors the context structure! The context indicates revenue $R$ as 12 times adult tickets $a$ plus 8 times student tickets $s$, so $R = 12a + 8s$; when $s=20$ is fixed, graphing $R$ versus $a$ means x-axis for $a$ (Adult tickets, tickets) and y-axis for $R$ (Revenue, dollars). Choice A correctly creates the equation $R = 12a + 8s$ and sets up the graph with proper variables and labels for fixed $s=20$. Choice B incorrectly adds the rates before multiplying and swaps axes, which would plot dependent on independent incorrectly; instead, ensure the dependent variable (what you're solving for) goes on the y-axis. Equation creation framework: (1) Define your variables clearly—'Let x = [exactly what it represents] in [units]'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Proper graph setup requires three things: (1) axis labels that include both variable name AND units ('Time (hours)' not just 't'), (2) scale with intervals fitting your data range (if values go 0-100, mark every 10 or 20, not every 1 or every 100), (3) clear indication of what each axis represents. A well-labeled graph communicates information immediately—anyone should be able to read your graph without additional explanation!
9. A streaming service charges $9 per month plus $2 per movie rented. Let $m$ be the number of months, $n$ be the number of movies, and $C$ be total cost in dollars. Create an equation representing the total cost in terms of $m$ and $n$.
   Answer: $C=9m+2n$ (where $C$ is dollars, $m$ is months, $n$ is movies) This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For '$9 per month plus $2 per movie,' the dependent quantity is cost (C), independent quantities are months (m) and movies (n), and the relationship is additive with rates: C = 9m + 2n. The equation structure mirrors the context structure! The phrase '$9 per month' translates to 9m (9 times the number of months), and '$2 per movie' translates to 2n (2 times the number of movies), with 'plus' meaning we add these components: C = 9m + 2n. Choice A correctly creates the equation C = 9m + 2n and properly identifies that C is in dollars, m is months, and n is movies. Choice C incorrectly swaps the coefficients to C = 9n + 2m, which would mean $9 per movie and $2 per month—always match each coefficient to its correct variable based on the context! Equation creation framework: (1) Define your variables clearly—'Let m = number of months, n = number of movies rented, C = total cost in dollars'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, (3) Build the equation piece by piece: $9 per month → 9m, $2 per movie → 2n, plus → +, giving C = 9m + 2n, (4) Verify with a test: for 2 months and 3 movies, C = 9(2) + 2(3) = 18 + 6 = $24, which makes sense!
10. A rectangular garden has a perimeter of 120 feet. If the length is represented by $l$ and the width by $w$, and the area must be at least 800 square feet, which system of equations and inequalities correctly models this situation?
    Answer: $2l + 2w = 120$ and $lw \geq 800$ When you encounter geometry word problems involving perimeter and area constraints, you need to carefully translate each condition into mathematical language. For a rectangle's perimeter of 120 feet, remember that perimeter equals the sum of all four sides: $2l + 2w = 120$. This is your first equation. The area constraint states the area "must be at least 800 square feet," which translates to $lw \geq 800$ since area equals length times width. Choice D correctly captures both relationships: $2l + 2w = 120$ for the perimeter and $lw \geq 800$ for the area constraint. Choice A makes a critical error by writing $l + w = 60$ instead of $2l + 2w = 120$. While algebraically equivalent (you can divide the correct equation by 2), the problem asks for the system that "correctly models this situation," meaning it should directly reflect the given information. Additionally, it incorrectly doubles the area constraint to $2lw \geq 800$. Choice B compounds the perimeter error by using $l + w = 120$, which would mean the length plus width equals 120 feet—impossible since that's the entire perimeter. However, it correctly states the area inequality. Choice C correctly identifies the perimeter equation but completely misunderstands area, writing $l + w \geq 800$ instead of $lw \geq 800$. This confuses the linear sum with the multiplicative area formula. Remember: Always distinguish between perimeter (sum of sides) and area (length × width) formulas. Also, pay attention to whether constraints use "equals," "at least," or "at most" to choose between equations and inequalities.
11. A rectangle has length $(x+5)$ meters and width $(x-2)$ meters. Create an equation in two variables for the area $A$ (in square meters) in terms of $x$, then select appropriate axes labels and a reasonable scale to graph $A$ versus $x$ for $2 \le x \le 10$.
    Answer: Equation: $A = (x+5)(x-2) = x^2 + 3x - 10$; x-axis: $x$ (m); y-axis: Area $A$ (m$^2$); scale: x 2–10 by 1, y 0–130 by 10 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For a rectangle with length (x+5) and width (x-2), area equals length times width, so A = (x+5)(x-2). The equation structure mirrors the geometric relationship! To expand this product, we use FOIL: First terms x·x = x², Outer terms x·(-2) = -2x, Inner terms 5·x = 5x, Last terms 5·(-2) = -10. Combining like terms: A = x² + (-2x + 5x) + (-10) = x² + 3x - 10. Choice A correctly creates the equation A = x² + 3x - 10 with proper axis labels including units (x in meters, Area in square meters) and a reasonable scale for the given domain. Choice B incorrectly adds the dimensions instead of multiplying (perimeter formula, not area), Choice C has a sign error in the expansion (should be +3x, not -3x), and Choice D divides instead of multiplying. Equation creation framework: (1) Define your variables clearly—'Let x = the variable dimension in meters'—being specific prevents confusion, (2) Identify the mathematical structure from context: area of rectangle = length × width, (3) Build the equation by substituting the given expressions: A = (x+5)(x-2), (4) Verify with a test value: when x = 3, length = 8m, width = 1m, so area = 8m². Using our equation: A = 3² + 3(3) - 10 = 9 + 9 - 10 = 8m². Perfect match!
12. A rectangle has length $(x+5)$ and width $(x-2)$, where $x$ is in inches. Write an equation for the area $A$ (square inches) in terms of $x$, and choose appropriate axes labels and a reasonable scale to graph $A$ for $0\le x\le 10$.
    Answer: Equation: $A=(x+5)(x-2)=x^2+3x-10$. Graph setup: x-axis x (inches), y-axis Area (in$^2$); x-scale 0 to 10 by 1, y-scale -20 to 140 by 20. This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For a rectangle with length (x+5) and width (x-2), area = length × width = (x+5)(x-2). Using FOIL: x² - 2x + 5x - 10 = x² + 3x - 10. The equation structure mirrors the context structure! Area of a rectangle is always length times width. With length = (x+5) and width = (x-2), we multiply: A = (x+5)(x-2). Using the FOIL method: First: x·x = x², Outer: x·(-2) = -2x, Inner: 5·x = 5x, Last: 5·(-2) = -10. Combining: A = x² - 2x + 5x - 10 = x² + 3x - 10. For the graph range, when x = 0: A = -10; when x = 10: A = 100 + 30 - 10 = 120. The y-scale from -20 to 140 accommodates this range. Choice A correctly creates the equation A = x² + 3x - 10 with proper graph setup: x-axis 'x (inches)', y-axis 'Area (in²)', and appropriate scales for the parabola. Choice C has the wrong coefficient for the x term (-3x instead of +3x), resulting from a sign error in the FOIL process. Equation creation framework: (1) Define your variables clearly—'Let x = the variable dimension in inches, A = area in square inches'—being specific prevents confusion, (2) Identify the mathematical structure: area of rectangle = length × width = (x+5)(x-2), requiring binomial multiplication, (3) Build the equation using FOIL: (x+5)(x-2) = x² - 2x + 5x - 10 = x² + 3x - 10, (4) Verify: when x = 5, length = 10, width = 3, so A = 30, and our formula gives 25 + 15 - 10 = 30 ✓. Graph scale decision process: (1) Find your data range—calculate A for x = 0, 2, 5, 10: A(0) = -10, A(2) = 4 + 6 - 10 = 0, A(5) = 30, A(10) = 120, (2) Y-values range from -10 to 120, so use -20 to 140 for margin, (3) Range of 160, divide by 8 to get 20, (4) Mark intervals: -20, 0, 20, 40, ..., 140. This clearly shows the parabola!
13. A bike rental shop charges a one-time sign-up fee of $12 plus $4 per hour of rental. Write an equation for the total cost $C$ (dollars) in terms of hours rented $h$, and choose appropriate axes labels and a reasonable scale to graph the relationship for $0\le h\le 10$.
    Answer: Equation: $C=12+4h$. Graph setup: x-axis Hours (h), y-axis Cost (dollars); x-scale 0 to 10 by 1, y-scale 0 to 60 by 5. This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $12 sign-up fee plus $4 per hour,' the dependent quantity is cost (C), independent quantity is hours (h), and the relationship is additive with a fixed fee plus a rate: C = 12 + 4h. The equation structure mirrors the context structure! In this bike rental problem, we have a fixed cost of $12 (the sign-up fee) plus a variable cost of $4 per hour. This translates directly to C = 12 + 4h, where the 12 represents the y-intercept (cost when h = 0) and 4 represents the slope (rate of change). For the graph, hours (h) is the independent variable on the x-axis, cost (C) is dependent on the y-axis, and with h ranging from 0 to 10, costs range from $12 to $52, making a y-scale of 0 to 60 by 5 appropriate. Choice A correctly creates the equation C = 12 + 4h with proper graph setup: x-axis labeled 'Hours (h)', y-axis labeled 'Cost (dollars)', and reasonable scales that show all data clearly. Choice B incorrectly multiplies 12 by h instead of 4, reversing the coefficients and also swaps the axes, putting the dependent variable (Cost) on the x-axis which violates convention. Equation creation framework: (1) Define your variables clearly—'Let h = hours rented, C = total cost in dollars'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'plus' means add, '$4 per hour' means multiply 4 by h, (3) Build the equation piece by piece matching each phrase in the context: 'sign-up fee of $12' → 12, 'plus' → +, '$4 per hour' → 4h, giving C = 12 + 4h, (4) Verify with a test value: for h = 5 hours, C = 12 + 4(5) = 32 dollars, which makes sense! Graph scale decision process: (1) Find your data range—h goes from 0 to 10, so C goes from 12 to 52, (2) For the y-axis, we need to show 12 to 52, so use 0 to 60 to have some margin, (3) Divide 60 by about 10 to get interval size of 6, round to nice number 5, (4) Mark intervals: 0, 5, 10, 15, ..., 60. Clean, readable, shows all data!
14. A concert venue charges $18 per ticket and has a fixed rental cost of $250 for the night. Let $q$ be the number of tickets sold. Revenue is $R=18q$ and cost is $C=250+6q$ (staffing depends on attendance). Create an equation for profit $P$ in terms of $q$, and choose appropriate axes labels and a reasonable scale to graph profit for $0 \le q \le 100$.
    Answer: Equation: $P = R - C = 18q - (250+6q) = 12q - 250$; x-axis: Tickets sold $q$ (tickets); y-axis: Profit $P$ ($); scale: x 0–100 by 10, y -300–1000 by 100 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For profit calculations, the fundamental relationship is Profit = Revenue - Cost. Here, Revenue R = 18q (price per ticket × quantity), Cost C = 250 + 6q (fixed cost + variable cost), so P = 18q - (250 + 6q) = 18q - 250 - 6q = 12q - 250. The equation structure mirrors the business relationship! Breaking down the profit calculation: Revenue from q tickets at $18 each is 18q. Total costs include fixed rental ($250) plus staffing costs that depend on attendance ($6 per ticket sold, so 6q). Profit = Revenue - Cost = 18q - (250 + 6q) = 18q - 250 - 6q = 12q - 250. Choice A correctly creates the equation P = 12q - 250 with proper axis labels and a scale that shows both negative profits (losses when q is small) and positive profits. Choice B incorrectly subtracts revenue from cost (should be revenue minus cost), Choice C has an arithmetic error (18q - 6q = 12q, not 24q), and Choice D adds instead of subtracting costs. Graph scale decision process: (1) Find your data range—when q = 0, P = -250; when q = 100, P = 12(100) - 250 = 950, (2) Y-axis needs to show from -300 to 1000, (3) Using intervals of 100 gives clean markings, (4) X-axis from 0 to 100 by 10s shows all relevant ticket quantities. This scale clearly shows the break-even point where profit crosses from negative to positive!
15. A cylindrical tank has volume $V=\pi r^2 h$, where $r$ is radius (meters) and $h$ is height (meters). If the height is fixed at $h=3$ meters, what two-variable equation relates $V$ and $r$, and what axes labels and a reasonable scale would you use to graph it for $0\le r\le 5$?
    Answer: Equation: $V=3\pi r^2$; x-axis: Radius (m); y-axis: Volume (m$^3$); scale: x 0–5 by 1, y 0–250 by 50 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For the cylinder volume formula V = πr²h with fixed height h = 3, we substitute to get V = πr²(3) = 3πr². The equation structure comes from substituting the constant! With h = 3 meters fixed, the volume formula V = πr²h becomes V = πr²(3) = 3πr², showing volume is proportional to the square of the radius. Choice A correctly creates the equation V = 3πr² with proper axis labels Radius (m) and Volume (m³), and scale x 0-5 by 1, y 0-250 by 50 to show the parabolic relationship. Choice D incorrectly uses V = πr³, which would be missing the factor of 3 from the height and incorrectly cubing the radius—cylinder volume uses r squared, not r cubed! Equation creation framework: (1) Start with the general formula: V = πr²h for cylinder volume, (2) Substitute the given constant: with h = 3, we get V = πr²(3), (3) Simplify by putting the coefficient first: V = 3πr², (4) Verify dimensions: r² gives area, times h gives volume ✓. Graph scale decision process: at r = 5, V = 3π(25) = 75π ≈ 236 m³, so y-axis to 250 by 50 captures this nicely!
16. A bike rental shop charges a one-time service fee of $12 plus $5 per hour. 1) Write an equation for the total cost $C$ (dollars) in terms of hours rented $h$ (hours). 2) Then choose the best axes labels and reasonable scales to graph the relationship for $0\le h\le 10$. Which choice gives both the correct equation and an appropriate graph setup?
    Answer: Equation: $C=12+5h$; Graph: x-axis Time (hours), y-axis Cost (dollars); x-scale 0 to 10 by 1, y-scale 0 to 70 by 10 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost (C), independent quantities are hours (h) and items (n), and the relationship is additive with rates: C = 40h + 15n. The equation structure mirrors the context structure! The context says a flat fee of $12 plus $5 per hour, so the dependent variable is cost C, independent is hours h, and the relationship is C = 12 + 5h, which is linear; for graphing 0 to 10 hours, x-axis should be time in hours (independent), y-axis cost in dollars (dependent), with x-scale 0-10 by 1 fitting the range nicely, and y-scale 0-70 by 10 since max cost is 12 + 5*10 = 62, rounded up for readability. Choice C correctly creates the equation C=12+5h with proper variables and coefficients, and sets up the graph with accurate labels including units and scales that cover the data without crowding. A common distractor like choice D uses a fraction instead of multiplication, perhaps misreading 'per' as division, but remember 'per hour' means multiply by the rate, so correct it by building the equation phrase by phrase: flat fee adds 12, plus 5 times hours. Equation creation framework: (1) Define your variables clearly—'Let h = hours rented (hours), C = total cost (dollars)'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Graph scale decision process: (1) Find your data range—what are the minimum and maximum values you need to show for each axis? (2) Divide that range by 5-10 to get interval size, (3) Round to a 'nice' number: use 1, 2, 5, 10, 20, 50, 100, etc. (not 7 or 13!), (4) Mark intervals starting at 0 (or other logical point). Example: data from 0 to 80 → range is 80, divided by 8 gives 10, so mark every 10: 0, 10, 20, ..., 80. Clean, readable, shows all data!
17. A bike rental shop charges a one-time service fee of $12 plus $5 per hour. 1) Write an equation for the total cost $C$ (dollars) in terms of hours rented $h$ (hours). 2) Then choose the best axes labels and reasonable scales to graph the relationship for $0\le h\le 10$. Which choice gives both the correct equation and an appropriate graph setup?​
    Answer: Equation: $C=12+5h$; Graph: x-axis Time (hours), y-axis Cost (dollars); x-scale 0 to 10 by 1, y-scale 0 to 70 by 10 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost (C), independent quantities are hours (h) and items (n), and the relationship is additive with rates: C = 40h + 15n. The equation structure mirrors the context structure! The context says a flat fee of $12 plus $5 per hour, so the dependent variable is cost C, independent is hours h, and the relationship is C = 12 + 5h, which is linear; for graphing 0 to 10 hours, x-axis should be time in hours (independent), y-axis cost in dollars (dependent), with x-scale 0-10 by 1 fitting the range nicely, and y-scale 0-70 by 10 since max cost is 12 + 5*10 = 62, rounded up for readability. Choice C correctly creates the equation C=12+5h with proper variables and coefficients, and sets up the graph with accurate labels including units and scales that cover the data without crowding. A common distractor like choice D uses a fraction instead of multiplication, perhaps misreading 'per' as division, but remember 'per hour' means multiply by the rate, so correct it by building the equation phrase by phrase: flat fee adds 12, plus 5 times hours. Equation creation framework: (1) Define your variables clearly—'Let h = hours rented (hours), C = total cost (dollars)'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Graph scale decision process: (1) Find your data range—what are the minimum and maximum values you need to show for each axis? (2) Divide that range by 5-10 to get interval size, (3) Round to a 'nice' number: use 1, 2, 5, 10, 20, 50, 100, etc. (not 7 or 13!), (4) Mark intervals starting at 0 (or other logical point). Example: data from 0 to 80 → range is 80, divided by 8 gives 10, so mark every 10: 0, 10, 20, ..., 80. Clean, readable, shows all data!
18. A rectangular garden has a perimeter of 48 feet. Let $l$ be the length (ft) and $w$ be the width (ft). Write an equation relating $l$ and $w$, and choose appropriate axes labels and a reasonable scale to graph the relationship in the first quadrant.
    Answer: Equation: $2l + 2w = 48$ (equivalently $w = 24 - l$); x-axis: Length $l$ (ft); y-axis: Width $w$ (ft); scale: x 0–24 by 2, y 0–24 by 2 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For a rectangle with perimeter 48 feet, the perimeter formula is P = 2l + 2w, so 2l + 2w = 48. This can be simplified by dividing by 2 to get l + w = 24, or solved for w to get w = 24 - l. The equation structure mirrors the geometric relationship! The perimeter of a rectangle is the sum of all four sides: length + width + length + width = 2l + 2w. Setting this equal to 48 gives 2l + 2w = 48. We can express this in multiple equivalent forms: dividing by 2 gives l + w = 24, or solving for w gives w = 24 - l. Choice B correctly creates the equation 2l + 2w = 48 (equivalently w = 24 - l) with proper axis labels and a reasonable scale for the first quadrant. Choice A incorrectly uses the area formula (lw) instead of perimeter, Choice C has the wrong constant (should be 24, not 48 after simplifying), and Choice D uses subtraction instead of addition. Proper graph setup requires: (1) recognizing that if l + w = 24, then both l and w must be between 0 and 24 (in the first quadrant), (2) choosing scales that show this full range clearly, (3) using consistent intervals (every 2 feet works well for 0 to 24). The graph will be a straight line from (0, 24) to (24, 0), showing all possible length-width combinations for the given perimeter!
19. A delivery company charges $30 per hour for a driver plus $2 per mile traveled. Let $h$ be hours and $m$ be miles. 1) Create an equation for total cost $C$ (dollars) in terms of $h$ and $m$. 2) To graph on a 2D coordinate plane, you decide to fix $h=3$ hours and graph cost as a function of miles $m$ for $0\le m\le 100$. Which choice gives the correct equation and a correct graph setup for the $h=3$ case?
    Answer: Equation: $C=30h+2m$; If $h=3$, then $C=90+2m$; Graph: x-axis Miles (miles), y-axis Cost ($); x-scale 0 to 100 by 10, y-scale 0 to 300 by 50 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'cost is $40 per hour plus $15 per item,' the dependent quantity is cost (C), independent quantities are hours (h) and items (n), and the relationship is additive with rates: C = 40h + 15n. The equation structure mirrors the context structure! Here, cost C = 30h + 2m from $30 per hour plus $2 per mile; fixing h=3 gives C=90 + 2m, graphed with x-axis miles (miles) as independent, y-axis cost ($) as dependent, x-scale 0-100 by 10 for even spacing, y-scale 0-300 by 50 since max is 90 + 2*100 = 290. Choice A correctly creates the equation C=30h+2m with accurate coefficients, substitutes h=3 properly, and sets up the graph with fitting labels and scales. A common error in choice B swaps rates, perhaps mixing per-hour and per-mile, but correct by linking each 'per' phrase directly to its variable multiplier. Equation creation framework: (1) Define your variables clearly—'Let h = hours, m = miles, C = total cost (dollars)'—being specific prevents confusion, (2) Identify the mathematical structure from context language: 'per' means multiply (rate), 'plus' means add, 'times' or 'product' means multiply, 'percent' means exponential, (3) Build the equation piece by piece matching each phrase in the context, (4) Verify with a test value: does your equation give sensible output for a reasonable input? This catches setup errors before solving! Graph scale decision process: (1) Find your data range—what are the minimum and maximum values you need to show for each axis? (2) Divide that range by 5-10 to get interval size, (3) Round to a 'nice' number: use 1, 2, 5, 10, 20, 50, 100, etc. (not 7 or 13!), (4) Mark intervals starting at 0 (or other logical point). Example: data from 0 to 80 → range is 80, divided by 8 gives 10, so mark every 10: 0, 10, 20, ..., 80. Clean, readable, shows all data!
20. A bacteria culture starts with 500 bacteria and grows by 8% each hour. Write an equation for the population $P$ in terms of time $t$ (hours), and choose appropriate axes labels and a reasonable scale to graph the model for $0 \le t \le 10$.
    Answer: Equation: $P = 500(1.08)^t$; x-axis: Time $t$ (hours); y-axis: Population $P$ (bacteria); scale: x 0–10 by 1, y 0–1200 by 200 This question tests your ability to translate real-world relationships into mathematical equations with two or more variables and set up graphs with appropriate labels and scales to visualize these relationships. Creating equations from contexts requires identifying: (1) which quantities vary (your variables), (2) which depends on which (independent vs dependent), (3) the mathematical relationship connecting them (linear rate, quadratic area, exponential growth, etc.). For 'grows by 8% each hour,' this is exponential growth where the population is multiplied by 1.08 each hour (100% + 8% = 108% = 1.08). Starting with 500 bacteria gives P = 500$(1.08)^t$. The equation structure mirrors the growth pattern! Exponential growth means each hour the population is multiplied by the growth factor. Starting amount is 500, growth factor is 1.08 (not 0.08, which would mean shrinking to 8% of previous), and t represents the number of time periods. After 1 hour: P = 500(1.08)¹ = 540. After 2 hours: P = 500(1.08)² = 583.2. Choice C correctly creates the equation P = 500$(1.08)^t$ with proper axis labels and a reasonable scale showing the exponential curve. Choice A uses 0.08 instead of 1.08 (this would make the population shrink rapidly), Choice B uses linear growth instead of exponential, and Choice D incorrectly makes 500 the base instead of the initial value. Equation creation framework: (1) Define your variables clearly—'Let t = time in hours, P = population count'—being specific prevents confusion, (2) Identify the mathematical structure: 'grows by r% each period' means exponential with base (1 + r/100), (3) Build the equation: P = initial × (growth factor)^time = 500$(1.08)^t$, (4) Verify: at t = 0, P = 500(1)¹ = 500 ✓. Growth by 8% means multiply by 1.08, not 0.08!

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