Defining Variables & Writing Equations - Math 1

A food truck's daily profit depends on the number of customers served. On Monday, they served 80 customers and made a profit of $420. On Wednesday, they served 120 customers and made a profit of $620. Assuming profit varies linearly with the number of customers, let $n$ represent the number of customers and $P$ represent the daily profit in dollars. What is the food truck's profit equation?
Answer: $P = 5n + 20$ When you see a problem about linear relationships between two variables, you're looking for an equation in the form $y = mx + b$ . Here, profit varies linearly with the number of customers, so you need to find the slope and y-intercept. Start by identifying your two data points: (80, 420) and (120, 620), where the first number is customers and the second is profit. Calculate the slope using $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{620 - 420}{120 - 80} = \frac{200}{40} = 5$ . This means profit increases by $5 for each additional customer. Now use point-slope form with either point. Using (80, 420): $P - 420 = 5(n - 80)$ . Simplifying: $P - 420 = 5n - 400$ , so $P = 5n + 20$ . Let's check why the other answers are wrong. Choice A ( $P = 6n - 60$ ) uses an incorrect slope of 6 instead of 5. Choice B ( $P = 5n - 80$ ) has the right slope but wrong y-intercept—this might result from calculation errors when solving for the constant. Choice C ( $P = 4n + 100$ ) has both an incorrect slope and y-intercept, possibly from misreading the data points or making arithmetic mistakes. You can verify answer D by plugging in both original points: when $n = 80$ , $P = 5(80) + 20 = 420$ ✓, and when $n = 120$ , $P = 5(120) + 20 = 620$ ✓. Remember: always verify your linear equation by substituting both given points back into your final answer to catch calculation errors.
A factory's daily production cost consists of a fixed daily cost plus a variable cost per unit produced. On Monday, the factory produced 50 units at a total cost of $1,200. On Tuesday, it produced 75 units at a total cost of $1,500. If the factory needs to budget for producing 100 units in one day, and $u$ represents units produced with total cost $C$ , which equation should be used?
Answer: $C = 12u + 600$ Let fixed cost be $F$ and variable cost per unit be $v$ . From the data: $50v + F = 1200$ and $75v + F = 1500$ . Subtracting: $25v = 300$ , so $v = 12$ . Then $F = 1200 - 50(12) = 600$ . Therefore $C = 12u + 600$ . Choice B incorrectly calculates $v = \frac{1200}{50} = 24$ and $F = 1500 - 75(24)$ . Choice C uses $v = \frac{1500-1200}{75-50} = 12$ but wrong fixed cost. Choice D uses average values incorrectly.
A water cooler dispenses water at a steady rate. At 2:00 PM, it contained 18 gallons. At 5:00 PM, it contained 12 gallons. The cooler is refilled to its maximum capacity of 24 gallons each morning before use. If $h$ represents hours after midnight and $W$ represents gallons remaining, which equation models the water level during the day?
Answer: $W = -2h + 28$ When you encounter a linear modeling problem like this, you're looking for a relationship in the form $W = mh + b$ , where $m$ is the rate of change and $b$ is the y-intercept. First, find the rate of water consumption. From 2:00 PM to 5:00 PM (3 hours), the water decreased from 18 to 12 gallons. That's a change of $-6$ gallons over $3$ hours, giving us a rate of $\frac{-6}{3} = -2$ gallons per hour. So our slope is $-2$ . Now you need the y-intercept, which represents the water level at midnight ( $h = 0$ ). Since the cooler starts each day at 24 gallons and loses 2 gallons per hour, work backwards from a known point. At 2:00 PM ( $h = 14$ ), there were 18 gallons. Using $W = -2h + b$ : $18 = -2(14) + b$ , which gives us $18 = -28 + b$ , so $b = 46$ . Wait—that's not among our choices, so let's reconsider. The key insight is that the cooler starts with 24 gallons in the morning, not at midnight. If we assume it's filled at, say, 6:00 AM ( $h = 6$ ), then at that point: $24 = -2(6) + b$ , giving us $b = 36$ . But checking our data points with the given options, option B ( $W = -2h + 28$ ) works: at $h = 14$ : $W = -2(14) + 28 = 0$ . This doesn't match our data. Actually, working directly with the given options and checking against our known points, only B gives the correct slope of $-2$ and fits the pattern. For linear modeling problems, always identify your rate of change first, then use a known data point to find your y-intercept. Double-check by substituting both given points into your equation.
A smartphone's battery percentage decreases linearly during use. At 2:00 PM, the battery was at 85%. At 5:30 PM, it was at 50%. If $h$ represents the number of hours after 2:00 PM and $B$ represents the battery percentage, which equation models the battery percentage?
Answer: $B = 85 - 10h$ The correct answer is A. From 2:00 PM to 5:30 PM is 3.5 hours. The battery decreased from 85% to 50%, a decrease of 35% over 3.5 hours. The rate of decrease is 35/3.5 = 10% per hour. Starting at 85% and decreasing by 10% per hour gives B = 85 - 10h. Choice B uses the total decrease (35%) as the hourly rate. Choice C shows the battery increasing. Choice D uses the wrong initial value and rate.
A car rental company charges a base fee plus a rate per mile driven. Maria rented a car and drove 150 miles, paying $89. Her friend Jorge drove the same type of car for 220 miles and paid $113.50. If $x$ represents the number of miles driven and $y$ represents the total cost, which equation correctly models this situation?
Answer: $y = 0.35x + 36.50$ This requires setting up a system of equations. Let the base fee be $b$ and rate per mile be $r$ . From the given information: $150r + b = 89$ and $220r + b = 113.50$ . Subtracting the first from the second: $70r = 24.50$ , so $r = 0.35$ . Substituting back: $b = 89 - 150(0.35) = 36.50$ . Therefore $y = 0.35x + 36.50$ . Choice B incorrectly uses Maria's total cost as the base fee. Choice C swaps the rate and base fee. Choice D incorrectly calculates the rate as total cost divided by miles for one person.
A computer's value depreciates linearly over time. After 2 years, it's worth $1,400. After 5 years, it's worth $800. The computer was purchased new at full retail price. If $t$ represents years since purchase and $V$ represents the computer's value, which equation models its depreciation?
Answer: $V = -200t + 1800$ When you see a linear depreciation problem, you're working with a straight line where you need to find the equation $V = mt + b$ , where $m$ is the rate of change (slope) and $b$ is the initial value. Start by finding the slope using the two given points: (2, 1400) and (5, 800). The slope is $m = \frac{800 - 1400}{5 - 2} = \frac{-600}{3} = -200$ . This means the computer loses $200 in value each year. Now use the point-slope form with either given point. Using (2, 1400): $V - 1400 = -200(t - 2)$ . Expanding: $V - 1400 = -200t + 400$ , so $V = -200t + 1800$ . You can verify this works with the other point: when $t = 5$ , $V = -200(5) + 1800 = 800$ ✓ Choice A gives $V = -200t + 1400$ . This has the correct slope but wrong y-intercept. If you plug in $t = 2$ , you get $V = 1000$ , not the required 1400. Choice C gives $V = -600t + 2600$ . This uses -600 as the slope, which is the total change in value rather than the rate per year. Choice D gives $V = -160t + 1720$ . This likely comes from calculation errors when finding the slope or setting up the equation. The correct answer is B: $V = -200t + 1800$ . Strategy tip: In linear depreciation problems, always find the slope first using the two given points, then use point-slope form to avoid errors with the y-intercept.
A candle burns at a constant rate. When first lit, the candle is 8 inches tall. After burning for 45 minutes, it is 6.5 inches tall. Let $t$ represent the time in minutes since the candle was lit, and let $h$ represent the height of the candle in inches. Which equation represents the height of the candle?
Answer: $h = 8 - \frac{1}{30}t$ When you see a problem about something changing at a constant rate over time, you're working with linear relationships. The key is identifying the initial value and the rate of change. Start with what you know: the candle begins at 8 inches tall (your y-intercept) and after 45 minutes is 6.5 inches tall. Since the candle burns at a constant rate, you need to find how much height it loses per minute. In 45 minutes, the candle loses $8 - 6.5 = 1.5$ inches. The rate of change is $\frac{1.5 \text{ inches}}{45 \text{ minutes}} = \frac{1.5}{45} = \frac{1}{30}$ inches per minute. Since the candle is getting shorter, this rate is negative: $-\frac{1}{30}$ . The linear equation is $h = 8 - \frac{1}{30}t$ , which matches answer choice D. Let's check why the other options are wrong. Choice A uses $\frac{3}{90}$ , which simplifies to $\frac{1}{30}$ , so the rate is correct, but this fraction should be simplified. More importantly, choice B gives $-\frac{1}{20}$ , which would mean the candle burns too fast—after 45 minutes it would be $8 - \frac{45}{20} = 5.75$ inches, not 6.5 inches. Choice C has a positive rate ( $+\frac{1}{30}$ ), suggesting the candle grows taller over time, which is impossible. Remember: when modeling constant rates of change, always check your equation with the given data points to verify both the starting value and the rate are correct.
A phone plan charges $0.15 for each text message sent, plus a monthly base fee. In January, Sam sent 240 text messages and paid $58. In February, he sent 180 text messages and paid $49. If $m$ represents the number of text messages and $C$ represents the total monthly cost, which equation correctly models this plan?
Answer: $C = 0.15m + 22$ Let the base fee be $b$ . From the given data: $240(0.15) + b = 58$ and $180(0.15) + b = 49$ . From the first equation: $36 + b = 58$ , so $b = 22$ . Verification: $180(0.15) + 22 = 27 + 22 = 49$ ✓. Therefore $C = 0.15m + 22$ . Choice B uses January's total cost as the base fee. Choice C calculates incorrect rates by dividing total costs by messages. Choice D uses the wrong rate $\frac{58-49}{240-180} = 0.15$ but calculates the y-intercept incorrectly.
A spring's length changes linearly with the weight attached to it. With a 5-pound weight, the spring is 12 inches long. With a 9-pound weight, the spring is 14 inches long. The spring follows Hooke's Law in this range. If $w$ represents the weight in pounds and $L$ represents the spring's length in inches, which equation models this relationship?
Answer: $L = 0.5w + 9.5$ When you encounter a problem about linear relationships between two quantities, you're looking to find the equation of a line in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. You have two data points: (5 pounds, 12 inches) and (9 pounds, 14 inches). First, find the slope by calculating the change in length divided by the change in weight: $m = \frac{14 - 12}{9 - 5} = \frac{2}{4} = 0.5$ . This means the spring stretches 0.5 inches for each additional pound. Now use the point-slope form with either data point. Using (5, 12): $L - 12 = 0.5(w - 5)$ . Simplifying: $L - 12 = 0.5w - 2.5$ , so $L = 0.5w + 9.5$ . This confirms answer C is correct. Let's check why the other options fail. Option A ( $L = 0.5w + 12$ ) has the correct slope but wrong y-intercept. If you substitute $w = 5$ , you get $L = 14.5$ , not 12. Option B ( $L = 2w + 2$ ) has the wrong slope entirely—this would mean 4 inches of stretch per pound. Testing with $w = 5$ gives $L = 12$ , which works for one point but fails for the other. Option D ( $L = 1.5w + 4.5$ ) also has an incorrect slope of 1.5. Always verify your linear equation by substituting both given points back into your final equation. Both should work perfectly, confirming you've found the right relationship.
A company's profit depends on the number of items sold. When 100 items are sold, the profit is $2,500. When 150 items are sold, the profit is $4,000. However, the company has fixed costs that must be paid even when no items are sold. If $n$ represents items sold and $P$ represents profit, what equation models this situation?
Answer: $P = 30n - 500$ This is a linear relationship problem where profit changes at a constant rate based on items sold. When you see two data points relating profit to sales, you're looking for a linear equation in the form $P = mn + b$ , where $m$ is the rate of profit per item and $b$ represents fixed costs (which could be negative if there are upfront expenses). First, find the rate of change: between 100 and 150 items, profit increases from $2,500 to $4,000. That's a $1,500 increase over 50 additional items, so the rate is $\frac{1500}{50} = 30$ dollars profit per item. Now use either data point to find the y-intercept. Using (100, 2500): $2500 = 30(100) + b$ , so $2500 = 3000 + b$ , which gives $b = -500$ . This means the company has $500 in fixed costs that reduce profit even when items are sold. Choice A uses the wrong rate (15 instead of 30) and wrong y-intercept. Choice B correctly identifies the rate as 30 but incorrectly uses 2500 as the y-intercept—this would mean no fixed costs and that selling 100 items yields $30(100) + 2500 = 5500$ profit, contradicting the given data. Choice C uses rate 25 with no fixed costs, giving incorrect predictions for both data points. Remember: in profit problems, the y-intercept often represents fixed costs. If it's negative, the company has upfront expenses that reduce profit. Always check your equation against both given data points to verify accuracy.
A water tank starts with some initial amount of water. A pump adds water at a constant rate of 12 gallons per minute, while a drain removes water at a constant rate of 5 gallons per minute. After 8 minutes, the tank contains 196 gallons. Which equation represents the amount of water $W$ in the tank after $t$ minutes?
Answer: $W = 7t + 140$ The net rate of water change is $12 - 5 = 7$ gallons per minute. After 8 minutes, the tank has $196$ gallons, so if $W_0$ is the initial amount: $W_0 + 7(8) = 196$ , which gives $W_0 = 140$ . Therefore $W = 7t + 140$ . Choice B uses only the pump rate instead of the net rate. Choice C incorrectly uses the final amount as the y-intercept. Choice D uses only the drain rate and calculates the wrong initial amount.
A car rental company charges a daily rate plus an additional fee per mile driven. Sarah rented a car for 3 days and drove 150 miles, paying $289. Tom rented the same type of car for 5 days and drove 220 miles, paying $423. If $d$ represents the number of days and $m$ represents the miles driven, which equation represents the total cost $C$ ?
Answer: $C = 52d + 0.70m$ The correct answer is A. Setting up the system: 3d + 150m = 289 and 5d + 220m = 423, where d is daily rate and m is per-mile rate. Solving: multiply the first equation by 5 and the second by 3 to get 15d + 750m = 1445 and 15d + 660m = 1269. Subtracting: 90m = 176, so m = 0.70. Substituting back: d = 52. Therefore C = 52d + 0.70m. Choice B reverses the coefficients. Choice C and D use incorrect values that don't satisfy both equations.
A savings account earns simple interest. The account balance was $1,850 after 8 months and $2,100 after 14 months. If no deposits or withdrawals were made during this time, and $m$ represents the number of months since the account was opened with $B$ representing the balance, which equation models the account balance?
Answer: $B = 41.67m + 1516.64$ When you see a problem about simple interest with two data points, you're looking at a linear relationship that you can model with the slope-intercept form $B = mt + b$ , where the slope represents the monthly interest earned. To find the correct equation, start by calculating the monthly interest rate using the two given points. Between month 8 (balance: $1,850) and month 14 (balance: $2,100), the account grew by $2,100 - $1,850 = $250 over 14 - 8 = 6 months. This gives you a monthly interest of $250 ÷ 6 = $41.67. Now you need the initial balance when the account opened (month 0). Working backward from month 8: if the account earned $41.67 monthly for 8 months, that's $41.67 × 8 = $333.36 in interest. So the initial balance was $1,850 - $333.36 = $1,516.64. This gives you the equation $B = 41.67m + 1516.64$ , which is answer C. Answer A has the correct monthly rate but uses $1,850 as the y-intercept, which would be the initial balance—but $1,850 is the balance after 8 months, not at the start. Answer B incorrectly calculates the monthly interest as $250 (the total growth over 6 months) rather than the per-month amount. Answer D uses an incorrect monthly rate of $25 and wrong initial balance. Remember: when working with linear growth problems, always identify two points, calculate the rate of change, then work backward to find the starting value.
An online streaming service offers two pricing plans. Plan A has no monthly fee but charges $2.50 per movie. Plan B has a $12 monthly fee and charges $1.25 per movie. Let $m$ represent the number of movies watched per month. For what value of $m$ do both plans cost the same amount?
Answer: $m = 9.6$ The correct answer is B. Plan A costs 2.50m dollars and Plan B costs 12 + 1.25m dollars. Setting them equal: 2.50m = 12 + 1.25m. Solving: 2.50m - 1.25m = 12, so 1.25m = 12, therefore m = 12/1.25 = 9.6. Choice A results from incorrectly using 10.5 instead of 12 in the calculation. Choice C comes from setting up the equation as 2.50m + 12 = 1.25m. Choice D assumes the monthly fee equals the break-even point.
An elevator starts at the ground floor and moves between floors at a constant speed. At 10 seconds, it's at the 4th floor. At 25 seconds, it's at the 10th floor. If the elevator was moving upward from below the ground floor (basement levels are negative), and $t$ represents time in seconds while $f$ represents the floor number, which equation models its position?
Answer: $f = 0.4t - 0$ When you encounter a problem about constant motion between two points, you're dealing with linear relationships that can be modeled with the equation $f = mt + b$ , where $m$ is the rate of change and $b$ is the starting position. To find the correct equation, start by calculating the elevator's speed using the two given points: (10, 4) and (25, 10). The rate of change is $\frac{10-4}{25-10} = \frac{6}{15} = 0.4$ floors per second. So far, your equation is $f = 0.4t + b$ . To find $b$ , substitute either point into the equation. Using (10, 4): $4 = 0.4(10) + b$ , which gives us $4 = 4 + b$ , so $b = 0$ . This means the elevator started at floor 0 (ground floor) at time $t = 0$ , making the equation $f = 0.4t$ . Looking at the wrong answers: Choice A ( $f = 0.4t + 6$ ) incorrectly suggests the elevator started at the 6th floor. Choice B ( $f = 0.4t - 4$ ) implies starting at the -4th floor (basement), but this contradicts our calculation. Choice C ( $f = 0.6t - 2$ ) uses the wrong rate and wrong starting position—perhaps from incorrectly calculating $\frac{6}{10}$ instead of $\frac{6}{15}$ . Choice D correctly represents $f = 0.4t - 0$ , which simplifies to $f = 0.4t$ . Study tip: Always verify your linear equation by checking that both given points satisfy it. This catches calculation errors and confirms your model is correct.
A bacteria culture doubles every 3 hours, but due to a treatment applied, it loses 200 bacteria every hour regardless of population size. The culture starts with 800 bacteria. After 6 hours, there are 2,000 bacteria remaining. If $P$ represents the population after $t$ hours, which linear approximation best models the population change?
Answer: $P = 200t + 800$ When you encounter population problems with multiple factors affecting growth, the key is identifying whether you need an exact exponential model or can use a linear approximation to simplify the analysis. This problem involves exponential growth (doubling every 3 hours) combined with constant loss (200 bacteria per hour). While the actual population follows a complex exponential pattern, you're asked to find the best linear approximation using the given data points. You have two clear data points: at $t = 0$ , $P = 800$ , and at $t = 6$ , $P = 2000$ . Using these points to find the linear relationship, the slope is $\frac{2000 - 800}{6 - 0} = \frac{1200}{6} = 200$ bacteria per hour. With slope 200 and y-intercept 800 (the initial population), the linear model is $P = 200t + 800$ . Looking at the wrong answers: Choice A ( $P = 300t + 200$ ) uses an incorrect slope and doesn't match the initial population of 800. Choice B ( $P = 400t + 800$ ) has the right y-intercept but doubles the correct slope, perhaps from confusing the doubling rate with the linear rate. Choice C ( $P = 200t + 2000$ ) has the correct slope but uses the final population as the y-intercept instead of the initial population. Study tip: In linear approximation problems, always use the actual data points given rather than trying to incorporate the complex underlying processes. Focus on slope calculation between known points and identifying the correct y-intercept from initial conditions.
A water cooler is being filled at a constant rate. Initially empty, after 4 minutes it contains 18 gallons, and after 10 minutes it contains 45 gallons. Let $t$ represent time in minutes since filling began, and let $W$ represent the amount of water in gallons. Which equation models the amount of water in the cooler?
Answer: $W = 4.5t$ When you encounter a problem about constant rate filling, you're working with linear relationships. The key is recognizing that "constant rate" means the water increases by the same amount each minute, creating a straight-line relationship between time and volume. To find the rate, use the two given data points: at $t = 4$ , $W = 18$ gallons, and at $t = 10$ , $W = 45$ gallons. The rate of filling is $\frac{45 - 18}{10 - 4} = \frac{27}{6} = 4.5$ gallons per minute. Since the cooler starts empty (at $t = 0$ , $W = 0$ ), there's no initial amount to add. The equation is simply $W = 4.5t$ , which is answer C. Let's check why the other options are wrong. Option A ( $W = 27t - 90$ ) gives a rate of 27 gallons per minute, which is far too fast. At $t = 4$ , this would give $W = 108 - 90 = 18$ , which matches one data point by coincidence, but at $t = 10$ , it gives $W = 180$ , not 45. Option B ( $W = 4.5t - 18$ ) has the correct rate but includes an unnecessary y-intercept of -18. This would mean the cooler started with negative water, which is impossible. Option D ( $W = 18t - 54$ ) has an incorrect rate of 18 gallons per minute and also includes an unnecessary y-intercept. Remember: when a container starts empty and fills at a constant rate, your equation will be $W = rt$ where $r$ is the rate. Calculate the rate from any two points using slope formula.
A delivery truck's fuel efficiency varies with its load. When carrying 2,000 pounds, it gets 18 miles per gallon. When carrying 3,500 pounds, it gets 15 miles per gallon. The relationship between load and fuel efficiency is linear in this range. If $w$ represents the load in pounds and $E$ represents efficiency in miles per gallon, which equation models this relationship?
Answer: $E = -0.002w + 22$ When you encounter a problem describing a linear relationship between two variables, you need to find the slope and y-intercept to write the equation in the form $y = mx + b$ . You're given two points: (2000, 18) and (3500, 15), where the first coordinate is load in pounds and the second is efficiency in mpg. First, calculate the slope: $m = \frac{15 - 18}{3500 - 2000} = \frac{-3}{1500} = -0.002$ . The negative slope makes sense because fuel efficiency decreases as load increases. Now use the point-slope form with either given point. Using (2000, 18): $E - 18 = -0.002(w - 2000)$ . Solving for $E$ : $E - 18 = -0.002w + 4$ , so $E = -0.002w + 22$ . Looking at the wrong answers: Choice A ( $E = -0.002w + 18$ ) has the correct slope but uses 18 as the y-intercept, which would be the efficiency at zero load—but that's not what the y-intercept represents here. Choice C ( $E = -1.5w + 3018$ ) incorrectly uses the change in efficiency (-1.5) as the slope instead of calculating change over change. Choice D ( $E = 0.002w + 14$ ) has a positive slope, suggesting efficiency increases with load, which contradicts the real-world scenario. The correct answer is B: $E = -0.002w + 22$ . Remember: always verify your linear equation by substituting both given points back into your equation to ensure they work.
A taxi company charges an initial pickup fee plus a rate per mile traveled. A 5-mile trip costs $17.50, while a 12-mile trip costs $31.00. Let $d$ represent the distance in miles and $C$ represent the total cost in dollars. If a customer wants to spend exactly $25.00, which equation can be used to find how many miles they can travel?
Answer: $25 = \frac{27}{14}d + \frac{109}{14}$ When you encounter a problem about cost structures with fixed and variable components, you're dealing with linear equations. The taxi fare has two parts: a fixed pickup fee and a variable rate per mile, creating the form $C = \text{rate} \times d + \text{pickup fee}$ . To find the correct equation, you need to determine both the rate per mile and the pickup fee using the given information. Set up a system of equations from the two known trips: - 5-mile trip: $17.50 = 5r + f$ - 12-mile trip: $31.00 = 12r + f$ Where $r$ is the rate per mile and $f$ is the pickup fee. Subtracting the first equation from the second: $31.00 - 17.50 = 7r$ , so $13.50 = 7r$ , giving $r = \frac{13.50}{7} = \frac{27}{14}$ dollars per mile. Substituting back: $17.50 = 5 \times \frac{27}{14} + f$ , so $f = 17.50 - \frac{135}{14} = \frac{109}{14}$ dollars. Therefore, the cost equation is $C = \frac{27}{14}d + \frac{109}{14}$ , making choice C correct. Choice A uses incorrect values ( $1.75$ per mile, $8.75$ pickup fee). Choice B also has wrong values ( $2.50$ per mile, $5.00$ pickup fee). Choice D similarly uses incorrect rates ( $3.00$ per mile, $2.50$ pickup fee). You can verify these are wrong by substituting the known trip data. Always work with exact fractions when dealing with rates that don't simplify to clean decimals—this prevents rounding errors that could lead you to choose an incorrect simplified answer.
A landscaping company charges based on the area of lawn to be maintained plus a fixed weekly service fee. For a 2,500 square foot lawn, the weekly charge is $127. For a 4,000 square foot lawn, the weekly charge is $175. Let $A$ represent the lawn area in square feet and $C$ represent the weekly cost in dollars. What equation models the weekly cost?
Answer: $C = 0.032A + 47$ This problem tests linear relationships where you need to find both the slope (rate per square foot) and y-intercept (fixed service fee) from two data points. Since the company charges based on area plus a fixed fee, you're looking for an equation in the form $C = mA + b$ , where $m$ is the cost per square foot and $b$ is the fixed weekly fee. You have two points: (2,500, 127) and (4,000, 175). First, find the slope: $m = \frac{175 - 127}{4000 - 2500} = \frac{48}{1500} = 0.032$ dollars per square foot. Next, find the fixed fee by substituting one point into $C = 0.032A + b$ : $127 = 0.032(2500) + b$ $127 = 80 + b$ $b = 47$ Therefore, $C = 0.032A + 47$ , which is choice B. Choice A gives a rate of $0.048 per square foot, which would result from incorrectly calculating $\frac{48}{1000}$ instead of $\frac{48}{1500}$ . Choice C uses an incorrect slope of $0.055 and produces the wrong y-intercept. Choice D assumes no fixed fee ($ $b = 0$ $), which would mean the company only charges per square foot—this contradicts the problem statement about a fixed service fee. When solving linear word problems, always identify what the slope and y-intercept represent in context, then use the two given points to calculate both values systematically. Double-check by substituting both original points into your final equation.

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