Setting Up Linear Equations - Accuplacer Qas

A plant's height increases linearly with time. On day 10 after planting, the plant was 15 cm tall. On day 25 after planting, it was 27 cm tall. Which equation can be used to find the plant's height $H$ (in cm) on day $d$ after planting?
Answer: $H = 0.8d + 7$ When you encounter a linear growth problem, you're working with a straight line relationship that can be expressed as $H = md + b$ , where $m$ is the slope (rate of change) and $b$ is the y-intercept. You have two data points: (10, 15) and (25, 27). First, find the slope using $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{27 - 15}{25 - 10} = \frac{12}{15} = 0.8$ cm per day. This means the plant grows 0.8 cm each day. Now use the point-slope form with either data point. Using (10, 15): $H = 0.8d + b$ , so $15 = 0.8(10) + b$ , which gives us $15 = 8 + b$ , therefore $b = 7$ . The equation is $H = 0.8d + 7$ . Let's check why the other answers are wrong. Answer A ( $H = 0.8d + 15$ ) has the correct slope but treats the height on day 10 as the y-intercept, which would mean the plant was 15 cm tall on day 0. Answer B ( $H = 1.25d - 0.5$ ) uses an incorrect slope of 1.25 instead of 0.8. Answer D ( $H = 12d - 105$ ) uses the height difference (12 cm) as the slope instead of dividing by the time difference. Always verify your linear equation by plugging in both given points to ensure they produce the correct heights. This catches calculation errors and confirms you've identified the right relationship between the variables.
A total of $8,000 is invested in two separate accounts. One account earns 3% simple interest annually, and the other earns 5% simple interest annually. The total interest earned from both accounts in one year is $340. If $x$ is the amount invested at 3%, which equation can be used to find $x$ ?
Answer: $0.03x + 0.05(8000 - x) = 340$ Let $x$ be the amount invested at 3%. The interest earned from this account is $0.03x$ . Since the total investment is $8,000, the amount invested in the other account at 5% is $8000 - x$ . The interest from this second account is $0.05(8000 - x)$ . The sum of the interest from both accounts is $340, so the equation is $0.03x + 0.05(8000 - x) = 340$ . * B incorrectly sets up the amount in the second account as $x - 8000$ , which would be a negative value. * C is an equation with two variables and does not use the information that the total investment is $8,000 to express $y$ in terms of $x$ . * D incorrectly sets the sum of the principal amounts equal to the interest earned, not the total principal.
The temperature in a laboratory freezer decreases linearly over time. At 2:00 PM, the temperature was 5°C. At 6:00 PM, the temperature was -15°C. If this cooling pattern continues, which equation gives the temperature $T$ (in degrees Celsius) at time $h$ hours after 2:00 PM?
Answer: $T = -5h + 5$ When you encounter a linear relationship problem, you're looking for an equation in the form $T = mh + b$ , where $m$ is the rate of change (slope) and $b$ is the initial value (y-intercept). Start by identifying your two data points: at $h = 0$ (2:00 PM), $T = 5°C$ , and at $h = 4$ (6:00 PM), $T = -15°C$ . Calculate the rate of change: $m = \frac{-15 - 5}{4 - 0} = \frac{-20}{4} = -5°C$ per hour. The negative slope makes sense since the temperature is decreasing. Since the temperature is 5°C when $h = 0$ , your y-intercept is 5. Therefore, the equation is $T = -5h + 5$ . Let's examine why the other options are incorrect. Choice A ( $T = -10h + 25$ ) has the wrong slope and y-intercept—plugging in $h = 0$ gives $T = 25°C$ , not 5°C. Choice B ( $T = 5h - 15$ ) has a positive slope, meaning temperature would increase over time, which contradicts the problem. Choice C ( $T = -20h + 45$ ) has too steep a slope and wrong y-intercept—at $h = 0$ , this gives $T = 45°C$ . You can verify answer D by checking both points: at $h = 0$ , $T = -5(0) + 5 = 5°C$ ✓, and at $h = 4$ , $T = -5(4) + 5 = -15°C$ ✓. Study tip: Always check your linear equation using both given points to confirm it's correct. This catches calculation errors and ensures your slope and y-intercept are accurate.
A company finds that the cost, $C$ , to produce $x$ widgets is given by $C = 3000 + 4x$ . The company wants to sell the widgets so that their profit is exactly $5000 when they sell 200 widgets. If $p$ is the selling price per widget, which equation can be used to find $p$ ?
Answer: $200p - (3000 + 4(200)) = 5000$ Profit is calculated as Total Revenue - Total Cost. The Total Revenue from selling 200 widgets at price $p$ is $200p$ . The Total Cost to produce 200 widgets is $C = 3000 + 4(200)$ . Setting the profit equal to $5000 gives the equation: Profit = Revenue - Cost, so $5000 = 200p - (3000 + 4(200))$ , which is equivalent to choice A. * B incorrectly adds the cost to the revenue instead of subtracting it. * C incorrectly structures the cost and revenue relationship. * D ignores the variable cost component ( $4x$ ) of the total cost.
An unknown temperature scale, Scale X, is linearly related to the Celsius scale. A temperature of 10°C corresponds to 40°X, and a temperature of 30°C corresponds to 90°X. If $c$ represents the temperature in Celsius, which equation correctly relates it to the temperature in Scale X, $x$ ?
Answer: $x = 2.5(c - 10) + 40$ A linear relationship can be described by the point-slope form $y - y_1 = m(x - x_1)$ . Let $c$ be the x-axis and $x$ be the y-axis. The slope $m$ is $\frac{\Delta x}{\Delta c} = \frac{90 - 40}{30 - 10} = \frac{50}{20} = 2.5$ . Using the point (10, 40), the equation is $x - 40 = 2.5(c - 10)$ , which rearranges to $x = 2.5(c - 10) + 40$ . * B incorrectly calculates the slope as 5. * C reverses the roles of $x$ and $c$ in the point-slope formula. * D uses the reciprocal of the correct slope.
The sum of three consecutive even integers is 138. If $n$ represents the smallest of these integers, which equation models this situation?
Answer: $3n + 6 = 138$ Consecutive even integers differ by 2. If the smallest integer is $n$ , the next consecutive even integer is $n + 2$ , and the third is $n + 4$ . Their sum is $n + (n + 2) + (n + 4) = 138$ . Combining like terms, we get $3n + 6 = 138$ . * A represents the sum of three consecutive integers, not consecutive even integers. * C represents the sum of three identical integers, not consecutive even integers. * D represents the sum of multiples of an integer, not consecutive even integers.
A taxi service charges a flat fee of $3.50 plus an additional $2.25 for each mile traveled. A passenger has at most $44.00 to spend on the ride. If $m$ represents the number of miles traveled, which of the following equations represents the exact number of miles the passenger can travel to spend their entire budget?
Answer: $2.25m + 3.50 = 44$ The total cost is the sum of the flat fee and the variable cost per mile. The flat fee is a constant $3.50. The variable cost is $2.25 multiplied by the number of miles, $m$ , which is (2.25m). Setting the sum of these two costs equal to the total budget of $44.00 gives the equation $2.25m + 3.50 = 44$ . * B incorrectly multiplies the number of miles by the flat fee and adds the per-mile charge as a constant. * C incorrectly adds the fee and per-mile charge before multiplying by the number of miles, implying the flat fee is also charged for every mile. * D incorrectly adds all quantities together without multiplying the per-mile charge by the number of miles.
A punch recipe calls for 3 parts cranberry juice to 5 parts ginger ale. If a total of 40 cups of punch is to be made, which equation can be used to find the number of cups of cranberry juice, $c$ , needed?
Answer: $c + \frac{5}{3}c = 40$ Let $c$ be the cups of cranberry juice and $g$ be the cups of ginger ale. The ratio is $c/g = 3/5$ , which can be written as $g = \frac{5}{3}c$ . The total amount is $c + g = 40$ . Substituting the expression for $g$ into the total equation gives $c + \frac{5}{3}c = 40$ . * B incorrectly reverses the ratio when solving for $g$ . * C incorrectly uses the ratio parts as multipliers for the volumes. * D incorrectly assumes cranberry juice is 3/5 of the total; it is 3/8 of the total (3 parts out of 3+5=8 total parts).
Two cars leave from the same city at the same time and travel in opposite directions. One car travels at an average speed of 55 miles per hour, and the other travels at an average speed of 65 miles per hour. Which equation can be used to find the time in hours, $t$ , it will take for the cars to be 480 miles apart?
Answer: $55t + 65t = 480$ When two objects travel in opposite directions, their relative speed is the sum of their individual speeds. The distance they are apart is this combined speed multiplied by time. The distance traveled by the first car is $55t$ and the distance by the second car is $65t$ . Since they are moving apart, the total distance between them is the sum of their individual distances: $55t + 65t = 480$ . * A represents the distance between two cars traveling in the same direction, where one is faster than the other. * C incorrectly multiplies the speeds instead of adding them. * D relates time and speed incorrectly, as if adding rates, not distances.
A coffee shop's daily profit depends linearly on the number of customers served. On Monday, they served 80 customers and made a profit of $340. On Tuesday, they served 120 customers and made a profit of $500. Which equation represents the daily profit $P$ when $n$ customers are served?
Answer: $P = 4n + 20$ When you encounter a problem stating that one quantity "depends linearly" on another, you're working with a linear function in the form $P = mn + b$ , where $m$ is the slope (rate of change) and $b$ is the y-intercept. You need to find both values using the given data points. Start by finding the slope using the two given points: (80, 340) and (120, 500). The slope represents how much profit changes per additional customer: $m = \frac{500 - 340}{120 - 80} = \frac{160}{40} = 4$ . So profit increases by $4 per customer. Now use the point-slope form with either point to find the y-intercept. Using (80, 340): $340 = 4(80) + b$ , so $340 = 320 + b$ , which gives $b = 20$ . Therefore, the equation is $P = 4n + 20$ . Choice A ( $P = 4.25n$ ) uses an incorrect slope calculation and assumes no fixed costs. Choice C ( $P = 160n - 12460$ ) incorrectly uses the profit difference (160) as the slope rather than dividing by the customer difference. Choice D ( $P = 4n + 340$ ) correctly identifies the slope but mistakenly uses one of the profit values as the y-intercept instead of calculating it properly. For linear relationship problems, always remember to calculate slope as "change in y over change in x," then substitute back into one of the original points to find the y-intercept. Don't assume the y-intercept equals one of your given data points.
A manufacturing plant has a fixed daily cost of $5,000 and a variable cost of $15 for each unit produced. The company sells each unit for $40. Which equation could be used to determine the number of units, $x$ , that must be produced and sold in a day to break even (i.e., for total revenue to equal total cost)?
Answer: $40x = 5000 + 15x$ The break-even point occurs when total revenue equals total cost. The total revenue from selling $x$ units is $40x$ . The total cost of producing $x$ units is the sum of the fixed cost and the variable cost, which is $5000 + 15x$ . Setting these two expressions equal gives the break-even equation: $40x = 5000 + 15x$ . * B incorrectly adds the revenue and variable cost, rather than equating revenue with total cost. * C is a correct rearrangement of the equation to find profit, but sets it to a negative value. The question asks for the setup where revenue equals cost. * D incorrectly places the revenue on the cost side of the equation.
A rental company charges for a leaf blower based on the equation $C = 8h + 15$ , where $C$ is the total cost in dollars and $h$ is the number of hours the blower is rented. A second company's price is described as a $25 flat fee plus $6 per hour. A customer wants to find the number of hours, $h$ , for which the cost from both companies is the same. Which equation should the customer solve?
Answer: $8h + 15 = 6h + 25$ To find when the costs are the same, we must set the expression for the first company's cost equal to the expression for the second company's cost. The first company's cost is given as $8h + 15$ . The second company's cost is a $25 flat fee plus $6 per hour, which is represented by $6h + 25$ . Setting them equal gives $8h + 15 = 6h + 25$ . * B incorrectly combines the hourly rates on one side and the flat fees on the other, rather than equating the total cost expressions. * C incorrectly associates the $25 fee with the number of hours for the second company. * D incorrectly subtracts the hourly cost from the flat fee for both companies.
Maya is twice as old as her cousin Ben. In 8 years, the sum of their ages will be 43. If $b$ represents Ben's current age, which equation can be used to find Ben's age?
Answer: $(2b + 8) + (b + 8) = 43$ Ben's current age is $b$ . Maya's current age is twice Ben's, so she is (2b). In 8 years, Ben's age will be $b + 8$ and Maya's age will be $2b + 8$ . The sum of their ages in 8 years will be 43. Therefore, the equation is $(b + 8) + (2b + 8) = 43$ . * A correctly states Maya's current age but only adds 8 years to Ben's age, not Maya's. * B adds 8 years only once, failing to account for both people aging by 8 years. * D incorrectly calculates Maya's future age by doubling Ben's future age, instead of adding 8 to her current age.
A rental car company charges a daily rate plus a fee per mile driven. The total cost for renting a car for one day and driving 120 miles is $89. The total cost for renting the same car for one day and driving 200 miles is $129. Which equation represents the total cost $C$ for driving $m$ miles in one day?
Answer: $C = 0.5m + 29$ Using points (120, 89) and (200, 129), the slope is (129-89)/(200-120) = 40/80 = 0.5. Using point (120, 89): C - 89 = 0.5(m - 120), so C = 0.5m + 29. Choice B incorrectly uses 89 as the y-intercept. Choice C switches the slope and y-intercept values. Choice D results from using an incorrect calculation method.
A delivery service charges based on package weight. Shipping a 3-pound package costs $12, and shipping a 7-pound package costs $20. If the cost structure is linear, which equation gives the shipping cost $C$ for a package weighing $w$ pounds?
Answer: $C = 2w + 6$ When you encounter a problem about linear cost structures, you're working with the equation of a line. You need to find the slope (rate of change) and y-intercept using the two given data points. Start by identifying your coordinate pairs: (3, 12) and (7, 20), where weight is x and cost is y. Calculate the slope using $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{20 - 12}{7 - 3} = \frac{8}{4} = 2$ . This means shipping costs increase by $2 per pound. Now use point-slope form with either point. Using (3, 12): $C - 12 = 2(w - 3)$ , which simplifies to $C = 2w + 6$ . You can verify this works for both points: when $w = 3$ , $C = 2(3) + 6 = 12$ ✓, and when $w = 7$ , $C = 2(7) + 6 = 20$ ✓. Choice A ( $C = 6w + 2$ ) uses the correct y-intercept but wrong slope—it would give costs of $20 and $44 for our test weights. Choice B ($ $C = 4w + 0$ $) has the wrong slope and assumes no base cost, yielding $12 and $28. Choice D ($ $C = 2.86w + 3.42$ $) appears to come from incorrect decimal calculations or a flawed approach to finding the line equation. Strategy tip: For linear relationship problems, always calculate the slope first using the two given points, then substitute back into one point to find the y-intercept. Double-check by testing both original data points in your final equation.
A salesperson earns a base salary of $1,200 per month plus a 5% commission on all sales exceeding $10,000. Last month, the salesperson's total sales were $S$ dollars, where $S > 10,000$ . Which equation represents the salesperson's total monthly earnings, $E$ ?
Answer: $E = 1200 + 0.05(S - 10000)$ The salesperson's earnings consist of the base salary ($1,200) and the commission. The commission is calculated only on the portion of sales exceeding $10,000. This amount is $S - 10,000$ . The commission is 5% of this amount, which is $0.05(S - 10,000)$ . Therefore, the total earnings are $E = 1200 + 0.05(S - 10000)$ . * A incorrectly applies the commission to the total sales amount, $S$ , rather than just the amount over $10,000. * C incorrectly calculates commission on the first $10,000 and then adds the total sales amount. * D incorrectly multiplies the base salary by the commission rate and adds the commissionable sales amount without the rate.
A chemist needs to create a 20-liter solution that is 35% acid. To do this, she will mix a solution that is 25% acid with another solution that is 50% acid. If $x$ represents the number of liters of the 25% acid solution, which equation can be used to determine the value of $x$ ?
Answer: $0.25x + 0.50(20 - x) = 0.35(20)$ The total amount of acid in the final mixture must equal the sum of the amounts of acid from the two initial solutions. The amount of acid from the 25% solution is $0.25x$ . Since the total volume is 20 liters, the volume of the 50% solution is $20 - x$ liters. The amount of acid from this solution is $0.50(20 - x)$ . The total amount of acid in the final 20-liter solution must be 35% of 20, which is $0.35(20)$ . Therefore, the equation is $0.25x + 0.50(20 - x) = 0.35(20)$ . * A incorrectly represents the volume of the 50% solution as $x - 20$ . * C equates the total volume of the solutions to the total amount of acid, which is incorrect. * D incorrectly calculates the amounts of acid and the final volume.
Jamal has a collection of dimes and quarters with a total value of $13.15. He has 11 more quarters than dimes. If $d$ represents the number of dimes, which equation represents the total value of the coins?
Answer: $0.10d + 0.25(d + 11) = 13.15$ The value of the dimes is $0.10d$ . The number of quarters is 11 more than the number of dimes, which is $d + 11$ . The value of the quarters is $0.25(d + 11)$ . The sum of these values is the total value, $13.15. Therefore, the equation is $0.10d + 0.25(d + 11) = 13.15$ . * B incorrectly subtracts 11, representing a situation where there are 11 fewer quarters than dimes. * C is an equation with two variables and does not use the relationship between the number of dimes and quarters. * D adds the number of coins, not their monetary value.
A cell phone plan costs $45 per month, which includes 5 gigabytes (GB) of data. There is an additional charge of $12 for each gigabyte of data used over the initial 5 GB. Last month's bill was $81. If $g$ represents the total number of gigabytes used, and $g > 5$ , which equation represents this scenario?
Answer: $45 + 12(g - 5) = 81$ The total bill consists of the flat fee ($45) and the overage charge. The overage charge applies only to the data used beyond the initial 5 GB. This amount of data is $g - 5$ . The cost for this extra data is $12 per gigabyte, or $12(g - 5)$ . The total bill is the sum of the flat fee and the overage charge, so the equation is $45 + 12(g - 5) = 81$ . * A incorrectly applies the overage charge to all gigabytes used, not just those over the limit. * C incorrectly calculates the number of overage gigabytes as $5 - g$ , which would be negative. * D incorrectly multiplies the flat fee by the number of gigabytes.
A spring stretches linearly under applied force. When a 12-newton force is applied, the spring stretches to 18 cm. When a 20-newton force is applied, the spring stretches to 26 cm. Which equation gives the stretched length $L$ (in cm) when a force of $F$ newtons is applied?
Answer: $L = F + 6$ When you see a problem about linear relationships, you're looking for a pattern where one variable changes at a constant rate relative to another. Here, the spring's length increases linearly with applied force. To find the equation, use the two given data points: (12 N, 18 cm) and (20 N, 26 cm). First, calculate the rate of change (slope): the length increases from 18 cm to 26 cm (a change of 8 cm) while the force increases from 12 N to 20 N (a change of 8 N). This gives a slope of 8÷8 = 1, meaning the spring stretches 1 cm per newton of additional force. Using the point-slope form with (12, 18): $L - 18 = 1(F - 12)$ , which simplifies to $L = F + 6$ . You can verify this works for both points: when $F = 12$ , $L = 12 + 6 = 18$ ✓, and when $F = 20$ , $L = 20 + 6 = 26$ ✓. Choice A ( $L = 1.5F - 8$ ) uses an incorrect slope of 1.5 instead of 1. Choice C ( $L = 8F + 18$ ) incorrectly uses the change in length (8) as the slope rather than the ratio of changes. Choice D ( $L = F + 18$ ) has the right slope but wrong y-intercept, likely from confusing one of the data points as the starting position. Always check your linear equation by substituting both given points back into your formula. If either point doesn't work, recalculate your slope and y-intercept.

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