SAT Math Quiz: Equations With Two Variables

What this quiz covers

This quiz focuses on Equations With Two Variables, giving you a quick way to practice the rules, question types, and explanations that matter most for SAT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $r$ decreases by 12
2. $r$ increases by 12 (correct answer)
3. $r$ increases by 320
4. $r$ decreases by 320

Explanation: Given p = 320 - r, we need to find how r changes when p decreases by 12. If p decreases by 12, then p_new = p - 12 = (320 - r) - 12 = 308 - r. Setting this equal to 320 - r_new, we get 308 - r = 320 - r_new, which gives us r_new = r + 12. Therefore, r increases by 12. The key insight is that in the equation p = 320 - r, p and r change in opposite directions. When working with inverse relationships, a decrease in one variable causes an increase in the other.

Question 2

1. $3x+5y=20$
2. $3x+5y=60$ (correct answer)
3. $5x+3y=60$
4. $\dfrac{3}{x}+\dfrac{5}{y}=60$

Explanation: This problem asks for an equation representing a flour constraint where x muffin batches use 3 cups each and y cookie batches use 5 cups each, totaling 60 cups. The total flour used is 3x + 5y, which must equal 60, giving us 3x + 5y = 60. Common errors include reversing the coefficients or using division instead of multiplication. When setting up constraint equations, multiply the number of items by the resource used per item, then sum to equal the total available.

Question 3

1. $1$
2. $2$ (correct answer)
3. $70$
4. $35$

Explanation: Given C = 120 + 35h, we need to find how h changes when C increases by 70 dollars. Since C is linear in h with coefficient 35, we have ΔC = 35 × Δh. Setting ΔC = 70: 70 = 35 × Δh, so Δh = 70/35 = 2 hours. The base cost of 120 doesn't affect the change. A common mistake is dividing by the wrong coefficient or including the constant term. For linear relationships, the change in output divided by the coefficient gives the change in input.

Question 4

1. $65$ (correct answer)
2. $79$
3. $72$
4. $-65$

Explanation: We need to calculate P using the formula P = 4c - 1i with c = 18 and i = 7. Substituting these values: P = 4(18) - 1(7) = 72 - 7 = 65. The coefficient 4 represents points gained per correct answer, while -1 represents the penalty per incorrect answer. A common error is to add instead of subtract the penalty points or to miscalculate the products. Always work through multiplication before addition/subtraction and pay attention to negative signs.

Question 5

1. $p=\dfrac{2.50n-C}{1.20}$
2. $p=\dfrac{C-2.50n}{1.20}$ (correct answer)
3. $p=\dfrac{C}{1.20}-2.50n$
4. $p=1.20(C-2.50n)$

Explanation: We need to solve for p in the equation C = 2.50n + 1.20p. First, subtract 2.50n from both sides to get C - 2.50n = 1.20p. Then divide both sides by 1.20 to isolate p, giving us p = (C - 2.50n)/1.20. A common mistake is to switch the order of subtraction in the numerator or to multiply by 1.20 instead of dividing. When isolating a variable with a coefficient, always divide by that coefficient as the final step.

Question 6

1. Dollars per mile (correct answer)
2. Miles per dollar
3. Total fare for 1 mile
4. Base fee in dollars

Explanation: In the fare equation F = 1.75m + 4.50, we need to interpret what 1.75 represents. Since m is the number of miles and F is in dollars, the coefficient 1.75 has units of dollars per mile - it's the rate charged for each mile traveled. The constant 4.50 represents the base fee charged regardless of distance. A common error is confusing the units or thinking 1.75 represents the base fee. In linear models, coefficients represent rates of change with respect to their variables.

Question 7

1. $(2,-3)$ (correct answer)
2. $(0,2)$
3. $(1,3)$
4. $(-1,-1)$

Explanation: The question asks which point satisfies y = -2x + 1, so substitute each x-coordinate and check whether the equation returns the listed y-value. For the point (2, -3): y = -2(2) + 1 = -4 + 1 = -3, which matches, so that point lies on the line. The point (1, 3) fails because -2(1) + 1 = -1, not 3; that value comes from ignoring the negative slope and computing 2(1) + 1. The point (0, 2) fails because -2(0) + 1 = 1, not 2, so it misidentifies the y-intercept, which is 1. The point (-1, -1) fails because -2(-1) + 1 = 2 + 1 = 3, not -1, an error from mishandling the double negative. Substituting carefully settles each case.

Question 8

1. $9$ (correct answer)
2. $0.18$
3. $35$
4. $53$

Explanation: We need to find how T changes when m increases by 50 in T = 35 + 0.18m. When m increases by 50, the new total is T_new = 35 + 0.18(m + 50) = 35 + 0.18m + 0.18(50) = 35 + 0.18m + 9. Comparing to the original T = 35 + 0.18m, we see T increases by 9. The fixed cost of 35 doesn't affect the change—only the per-mile rate of 0.18 matters. Students often forget to multiply the change by the coefficient or mistakenly include the constant term. In linear relationships, the change in output equals the coefficient times the change in input.

Question 9

1. $2a+5s=58$ (correct answer)
2. $5a+2s=58$
3. $2a-5s=58$
4. $\dfrac{2a}{5s}=58$

Explanation: We need to write an equation for 2 adult tickets at a dollars each and 5 student tickets at s dollars each totaling $58. The cost of 2 adult tickets is 2a, and the cost of 5 student tickets is 5s. The total is 2a + 5s = 58. Students often reverse the coefficients or use the wrong operation between terms. When translating word problems to equations, carefully identify what's being multiplied and ensure the units match.

Question 10

1. Increase by $2$
2. Decrease by $1$
3. Decrease by $2$ (correct answer)
4. Increase by $1$

Explanation: We need to determine how y changes when x increases by 1 in the constraint 2x + y = 9. If x increases by 1, the new equation is 2(x + 1) + y_new = 9, which gives 2x + 2 + y_new = 9. Since the original equation is 2x + y = 9, we can substitute to get 9 - y + 2 + y_new = 9, yielding y_new = y - 2. Therefore, y decreases by 2. The key insight is that to maintain the constraint, changes in variables must compensate for each other. When one variable in a linear constraint increases, analyze how the other must change to keep the equation balanced.

Question 11

1. $32$
2. $50$
3. $10$
4. $18$ (correct answer)

Explanation: This question asks how F changes when C increases by 10 in the formula $F = \frac{9}{5}C + 32$. Since F is a linear function of C, the change in F equals $\frac{9}{5}$ times the change in C. Calculate: $\Delta F = \frac{9}{5} \times 10 = \frac{90}{5} = 18$ degrees Fahrenheit. The constant 32 doesn't affect the change because it's the same before and after. A common error is including the constant term when calculating changes. For linear relationships, focus on the coefficient of the changing variable to find the rate of change.

Question 12

1. $p$ rises $40 per text
2. Base fee is $0.10
3. $t$ rises 0.10 per dollar
4. $p$ rises $0.10 per text (correct answer)

Explanation: In the phone plan equation $p = 40 + 0.10t$, we need to interpret the coefficient 0.10. This coefficient represents the rate of change of $p$ with respect to $t$, meaning for each additional text message (increase in $t$ by 1), the monthly cost $p$ increases by $0.10$. The 40 represents the base monthly fee before any texts. A common mistake is confusing which variable changes with respect to the other. In context problems, the coefficient of a variable tells you how much the dependent variable changes per unit increase in that independent variable.

Question 13

1. Increase by 6
2. Decrease by 6 (correct answer)
3. Increase by 2
4. Decrease by 2

Explanation: Given the equation 3x + 2y = 12, we need to find how y changes when x increases by 4. When x increases by 4, the term 3x increases by 3(4) = 12. To keep the sum equal to 12, the term 2y must decrease by 12, so y must decrease by 12/2 = 6. We can verify: if initially 3x + 2y = 12, then 3(x + 4) + 2(y - 6) = 3x + 12 + 2y - 12 = 3x + 2y = 12 ✓. A common error is forgetting to divide by the coefficient of y. The key is recognizing that changes must balance according to each variable's coefficient.

Question 14

1. $30$
2. $15$
3. $25$
4. $20$ (correct answer)

Explanation: This question asks for the monthly cost d in the model T = dm + 25, given T = 145 after m = 6 months. Substitute the values: 145 = d(6) + 25. Subtract 25 from both sides: 120 = 6d, then divide by 6: d = 20. This solves for d by isolating it in the linear equation. A key error could be forgetting to subtract 25, resulting in 145/6, which is not an integer choice. Another mistake might involve dividing by the wrong number. When working with cost models and two variables, substitute known quantities and solve step-by-step to reveal the rate variable's value.

Question 15

1. $P=600$, fixed cost $50$
2. $P=200$, fixed cost $-200$ (correct answer)
3. $P=600$, fixed cost $-200$
4. $P=200$, fixed cost $50$

Explanation: This question requires calculating profit P when n = 8 in P = 50n - 200 and identifying the fixed cost term. Substitute n = 8: 50 × 8 - 200 = 400 - 200 = 200, so profit is 200 dollars. The constant -200 represents the fixed cost, as it does not depend on n. Choices with P = 600 likely come from adding instead of subtracting 200, a sign error. Other errors might misidentify the coefficient 50 as fixed. When analyzing profit models, distinguish the constant as fixed components and the coefficient as variable per-unit contributions to understand cost-profit dynamics.

Question 16

1. $3x+2y=22$ (correct answer)
2. $2x+3y=22$
3. $3x-2y=22$
4. $\frac{3}{x}+\frac{2}{y}=22$

Explanation: This question requires finding the equation that models the relationship where 3 notebooks cost x dollars each and 2 pens cost y dollars each, totaling 22 dollars. The total cost is the sum of the costs for notebooks and pens, so set up the equation as 3x + 2y = 22. This directly represents the given quantities and total, matching choice A. A key error in choice B is swapping the coefficients, perhaps from miscounting the number of items. Choice C uses subtraction instead of addition, which might stem from incorrectly assuming a discount or refund, while choice D incorrectly uses reciprocals, unsuitable for total cost calculations. To handle such problems effectively, always multiply each item's quantity by its variable price and sum them to equal the total, emphasizing the additive relationship between variables.

Question 17

1. $4.25
2. $7.00 (correct answer)
3. $9.50
4. $2.50

Explanation: This question asks how much the total cost C changes when the miles traveled m increases by 4 miles in the linear model C = 2.50 + 1.75m. The equation is in slope-intercept form where 2.50 is the fixed booking fee and 1.75 is the per-mile rate, so changes in C depend only on changes in m multiplied by the slope. To find the change, multiply the increase in m by the coefficient of m: 1.75 × 4 = 7. A common error is including the flat fee in the calculation, but since it's constant, it doesn't affect the change in C. Another mistake might be misreading the rate or the increase in miles, leading to incorrect multiplication. In equations with two variables, focus on the coefficient of the independent variable to determine the rate of change in the dependent variable.

Question 18

1. $-10$ (correct answer)
2. $-5$
3. $5$
4. $10$

Explanation: This question asks how much y changes when x increases by 5 in 2x + y = 10. Rewrite as y = 10 - 2x; the slope is -2. Change in y: -2 × 5 = -10. A common error is not rewriting the equation to see the slope. Another mistake could be using a positive coefficient. In standard form equations, convert to slope-intercept to easily see the rate of change between variables.

Question 19

1. $s=60-\tfrac{3}{2}a$
2. $s=\tfrac{120-3a}{2}$ (correct answer)
3. $s=\tfrac{120+3a}{2}$
4. $s=\tfrac{3a-120}{2}$

Explanation: This question requires rewriting the equation $3a + 2s = 120$ to express s in terms of a. Begin by subtracting 3a from both sides to isolate the term with s: $2s = 120 - 3a$. Then, divide both sides by 2 to solve for s: $s = \tfrac{120 - 3a}{2}$. This form highlights the inverse relationship between a and s to maintain constant revenue. A key error might involve adding 3a instead of subtracting, leading to an incorrect positive term like in choice C. Another mistake could be dividing only part of the expression by 2. When manipulating equations with two variables, perform identical operations on both sides to isolate the target variable while preserving the relationship.

Question 20

1. 22
2. 13
3. 15
4. 19 (correct answer)

Explanation: This question requires substituting x = 4 into the linear equation $y = 3x + 7$ to find the corresponding y-value. Substituting x = 4 gives us $y = 3(4) + 7 = 12 + 7 = 19$. The calculation follows the order of operations: first multiply $3 \times 4 = 12$, then add 7 to get 19. A common error is forgetting to multiply before adding or making arithmetic mistakes during substitution. When evaluating linear equations, substitute the given value carefully and follow order of operations precisely.