PSAT 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 PSAT 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. $1$
2. $3$ (correct answer)
3. $7$
4. $19$

Explanation: The question asks for the value of s when f=4 in the equation 2f + s = 11. This is a linear equation with two variables, constraining the amounts of flour f and sugar s. Substitute f=4: 24 + s = 11, which is 8 + s = 11. Then, subtract 8 from both sides: s = 11 - 8 = 3. A common error is to multiply instead of adding or solving for f instead. Verify by plugging back: 24 + 3 = 8+3=11, correct. When given a value for one variable, substitute directly to find the other in two-variable equations.

Question 2

1. $n=\dfrac{T+s}{4}$
2. $n=\dfrac{T-s}{4}$ (correct answer)
3. $n=4(T-s)$
4. $n=T-4s$

Explanation: The question asks for the equation that solves for n in terms of T and s, given T = 4n + s. This equation models the total cost T as 4 times the price per notebook n plus shipping s. To solve for n, subtract s from both sides: T - s = 4n. Then, divide both sides by 4: n = (T - s)/4. A common error is to divide by something else or forget to subtract s first. Check by substituting back; if T=20, s=4, n=4, then 20=4*4 +4, yes, and (20-4)/4=4. Focus on inverse operations to isolate the desired variable in equations with multiple variables.

Question 3

1. $-1$
2. $5$
3. $10$ (correct answer)
4. $20$

Explanation: This problem involves y = (5/2)x - 1, where if x increases by 4, we need to find how y changes. The change in y equals the coefficient of x times the change in x: Δy = (5/2) × 4 = 5 × 2 = 10. The coefficient 5/2 represents the slope or rate of change. When x increases by 4, y increases by 10. A common error is making mistakes with fraction arithmetic or forgetting to multiply by the coefficient. In linear relationships, changes in the dependent variable equal the slope times changes in the independent variable.

Question 4

1. $y=2x-3$ (correct answer)
2. $y=\frac{1}{2}x+0$
3. $y=2x+3$
4. $y=\frac{1}{2}x-3$

Explanation: This problem asks for the equation of a line passing through (2,1) and (6,9). First, find the slope: m = (9-1)/(6-2) = 8/4 = 2. Using point-slope form with point (2,1): y - 1 = 2(x - 2), which gives y - 1 = 2x - 4, so y = 2x - 3. Let me verify with both points: at (2,1): y = 2(2) - 3 = 1 ✓; at (6,9): y = 2(6) - 3 = 9 ✓. A common error is making arithmetic mistakes in the slope calculation or when converting to slope-intercept form. When finding line equations, calculate slope first, then use point-slope form to find the equation.

Question 5

1. 25
2. 35 (correct answer)
3. 42
4. 60

Explanation: This is a linear equation application where you're given specific values and need to solve for an unknown variable. When you see a real-world scenario with a formula already provided, substitute the known values and solve algebraically. You're told the charity spends $72 total and buys 60 pens. Substitute these values into the equation $$0.50p + 1.20n = T$ $: $$0.50(60) + 1.20n = 72$$ First, calculate the cost of the pens: $$0.50 \times 60 = 30$$ So the equation becomes: $$30 + 1.20n = 72$$ Subtract 30 from both sides: $$1.20n = 42$$ Divide by 1.20: $$n = \frac{42}{1.20} = 35$$ The charity bought 35 notebooks. Looking at the wrong answers: Choice (A) 25 would result from incorrectly calculating $$\frac{30}{1.20}$$ instead of $$\frac{42}{1.20}$$, mixing up which dollar amount to divide. Choice (C) 42 comes from forgetting to divide by the notebook price—this is just the remaining dollars after buying pens, not the number of notebooks. Choice (D) 60 assumes they bought the same number of notebooks as pens, ignoring the price difference entirely. Strategy tip: In substitution problems, work methodically through each step and always check your answer makes sense. Here, 35 notebooks at $1.20 each plus 60 pens at $0.50 each should equal $72: $$35(1.20) + 60(0.50) = 42 + 30 = 72$$ ✓

Question 6

1. $s=60-\frac{3}{2}a$
2. $s=\frac{480-12a}{8}$ (correct answer)
3. $s=60-\frac{2}{3}a$
4. $s=\frac{12a-480}{8}$

Explanation: This problem asks to solve $12a + 8s = 480$ for s in terms of a. Starting with $12a + 8s = 480$, subtract 12a from both sides: $8s = 480 - 12a$. Then divide both sides by 8: $s = \frac{480 - 12a}{8}$. This can also be written as $s = 60 - 1.5a$, but the fraction form matches choice B exactly. A common error is incorrectly rearranging the terms or making sign errors when moving terms across the equal sign. When isolating a variable, perform the same operation to both sides of the equation systematically.

Question 7

1. $-4$ (correct answer)
2. $0$
3. $5$
4. $6$

Explanation: This problem asks for the y-intercept of a line passing through (0,-4) and (5,6). The y-intercept is the y-value when x = 0. Since one of the given points is (0,-4), the y-intercept is -4. The y-intercept can be read directly from a point where x = 0, without needing to find the full equation. A common error is confusing the y-intercept with the slope or using the wrong point. The y-intercept is simply the y-coordinate where the line crosses the y-axis (x = 0).

Question 8

1. Increase by 12 (correct answer)
2. Decrease by 12
3. Increase by 63
4. Decrease by 63

Explanation: This problem involves the constraint x + y = 75, where if y decreases by 12, we need to find how x must change to maintain the total. If y decreases by 12, then to keep x + y = 75, x must increase by 12. The constraint equation requires that any decrease in one variable be offset by an equal increase in the other variable. A common error is thinking both variables change in the same direction or getting the magnitude wrong. In constraint equations where variables sum to a constant, changes in the variables must be equal and opposite.

Question 9

1. $18$
2. $43$
3. $72$ (correct answer)
4. $100$

Explanation: This question asks how the total cost C changes when the number of months m increases by 4. The cost function is C = 25 + 18m, where 25 is the one-time sign-up fee and 18 is the monthly fee. When m increases by 4, the new cost becomes C_new = 25 + 18(m + 4) = 25 + 18m + 72. The change in cost is C_new - C = (25 + 18m + 72) - (25 + 18m) = 72 dollars. A common error is to only multiply the increase in months by the coefficient without considering the constant term, but since the constant 25 appears in both expressions, it cancels out. When dealing with linear functions, the change in output equals the rate of change times the change in input.

Question 10

1. $0$ (correct answer)
2. $2$
3. $6$
4. $12$

Explanation: This problem gives y = (-3/4)x + 6 and asks for y when x = 8. Substituting x = 8: y = (-3/4)(8) + 6 = -6 + 6 = 0. The calculation involves multiplying the fraction by 8: (-3/4) × 8 = -3 × 2 = -6, then adding 6 gives 0. A common error is making mistakes with fraction arithmetic or sign errors. When evaluating linear expressions with fractions, multiply carefully and follow the order of operations.

Question 11

1. Increases by $30$ (correct answer)
2. Increases by $45$
3. Decreases by $30$
4. Decreases by $45$

Explanation: This problem involves revenue R = 9a + 6c, where if c increases by 5 while a stays the same, we need to find how R changes. The change in R equals the coefficient of c times the change in c: ΔR = 6 × 5 = 30. Since the coefficient is positive and c increases, R increases by 30. The coefficient 6 represents the revenue per child ticket. A common error is using the wrong coefficient or forgetting that only one variable changes. When one variable in a linear expression changes while others stay constant, multiply the coefficient by the change in that variable.

Question 12

1. Increase by 10
2. Decrease by 10
3. Increase by 20
4. Decrease by 20 (correct answer)

Explanation: This problem involves the constraint 5x + 2y = 64, where if x increases by 4, we need to find how y must change to keep the equation satisfied. The change in the left side from increasing x by 4 is 5(4) = 20. To maintain equality, 2y must decrease by 20, so 2Δy = -20, which means Δy = -10. Therefore, y must decrease by 10. A common error is forgetting to account for the coefficients or getting the sign wrong. In constraint equations, when one term increases, another must decrease to maintain the constant sum.

Question 13

1. $230$ (correct answer)
2. $240$
3. $350$
4. $470$

Explanation: This problem gives the constraint r + v = 350 with v = 120, asking for r. Substituting v = 120: r + 120 = 350. Solving for r: r = 350 - 120 = 230. This represents a concert venue where the total number of regular and VIP seats is fixed. With 120 VIP seats, there are 230 regular seats. A common error is adding instead of subtracting or making arithmetic mistakes. In constraint equations where variables sum to a constant, one variable equals the constant minus the other.

Question 14

1. $x=4$
2. $x=8$ (correct answer)
3. $x=11$
4. $x=19$

Explanation: This problem gives x + 2y = 30 and asks for x when y = 11. Substituting y = 11: x + 2(11) = 30, which gives x + 22 = 30. Solving for x: x = 30 - 22 = 8. Let me verify: 8 + 2(11) = 8 + 22 = 30 ✓. This represents a packing constraint where large boxes hold twice as many items as small boxes. A common error is making arithmetic mistakes in substitution or subtraction. When solving constraint equations, substitute known values carefully and solve systematically.

Question 15

1. $x=9$
2. $x=11$
3. $x=15$
4. $x=21$ (correct answer)

Explanation: This problem gives $y = \frac{x - 6}{3}$ and asks for $x$ when $y = 5$. Substituting $y = 5$: $5 = \frac{x - 6}{3}$. Multiplying both sides by 3: $15 = x - 6$. Adding 6 to both sides: $x = 15 + 6 = 21$. Let me verify: $y = \frac{21 - 6}{3} = \frac{15}{3} = 5$ ✓. The equation can be solved by clearing the fraction first, then isolating the variable. A common error is making mistakes when multiplying by the denominator or combining terms. When solving equations with fractions, multiply both sides by the denominator to eliminate the fraction first.

Question 16

1. $140$
2. $240$
3. $260$ (correct answer)
4. $740$

Explanation: This problem gives W = 500 - 20t and asks for W when t = 12. Substituting t = 12: W = 500 - 20(12) = 500 - 240 = 260. This represents a water tank that starts with 500 liters and drains at 20 liters per minute. After 12 minutes, 260 liters remain. A common error is making arithmetic mistakes in the multiplication or subtraction. When evaluating linear expressions involving time, substitute the time value carefully and follow the order of operations.

Question 17

1. $t=40$
2. $t=50$
3. $t=60$ (correct answer)
4. $t=70$

Explanation: This problem gives C = 25 + 0.10t and asks for t when C = 31. Substituting C = 31: 31 = 25 + 0.10t. Solving for t: 31 - 25 = 0.10t, which gives 6 = 0.10t, so t = 6 ÷ 0.10 = 60. Let me verify: C = 25 + 0.10(60) = 25 + 6 = 31 ✓. The base plan costs $25, and each text message adds $0.10. A common error is making decimal arithmetic mistakes or forgetting to subtract the base cost first. When solving linear equations with decimals, isolate the variable term systematically.

Question 18

1. $-5$
2. $3$ (correct answer)
3. $9$
4. $27$

Explanation: This problem involves the equation y = (1/3)x - 5 and asks how y changes when x increases by 9. In a linear equation, the change in y equals the coefficient of x times the change in x: Δy = (1/3) × 9 = 3. The coefficient 1/3 represents the slope or rate of change. When x increases by 9, y increases by 3. A common error is forgetting to multiply by the coefficient or confusing the direction of change. In linear relationships of the form y = mx + b, when x changes by Δx, y changes by m × Δx.

Question 19

1. $+2$
2. $+8$ (correct answer)
3. $+10$
4. $-2$

Explanation: This problem involves P = 4q + 2h, where q increases by 3 and h decreases by 2. The net change in P is: ΔP = 4(3) + 2(-2) = 12 - 4 = 8. The increase from quizzes (+12 points) is partially offset by the decrease from homework (-4 points), resulting in a net increase of 8 points. A common error is forgetting to account for the signs or the coefficients of each variable. When multiple variables change in a linear expression, calculate each contribution separately, then sum them with proper signs.

Question 20

1. Base fee is $0.50$
2. Cost per download is $0.50$ (correct answer)
3. Downloads per dollar is $0.50$
4. Base fee is $12.50$

Explanation: This problem asks about the meaning of 0.50 in the equation c = 12 + 0.50d. In this linear cost model, 12 represents the base monthly fee (y-intercept), and 0.50 represents the additional cost per movie downloaded (slope). Each additional download increases the monthly cost by $0.50. The coefficient of the variable d represents the rate of change - the cost per download. A common error is confusing the coefficient with the base fee or misinterpreting the units. When interpreting linear models in context, the coefficient of the variable represents the rate of change per unit of that variable.