SAT Math Quiz: Equations With One Variable

What this quiz covers

This quiz focuses on Equations With One Variable, 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. $7.5$
2. $10$ (correct answer)
3. $12.5$
4. $\frac{25}{2}$

Explanation: We need to solve $2.5x - 7.5 = 0.5x + 12.5$ by first moving all x-terms to one side. Subtracting 0.5x from both sides gives $2x - 7.5 = 12.5$. Adding 7.5 to both sides yields $2x = 20$, so $x = 10$. A common mistake is making an arithmetic error when adding 7.5 + 12.5, perhaps getting 19 instead of 20, which would give $x = 9.5$. When working with decimals, take extra care with arithmetic and consider converting to fractions if that helps avoid errors.

Question 2

1. 5
2. $\frac{7}{30}$
3. $\frac{25}{7}$
4. $\frac{30}{7}$ (correct answer)

Explanation: Distribute and combine to get $10x - 5 = 3x + 25$, so $7x = 30$ and $x = \frac{30}{7}$. Other choices come from inverting the fraction, omitting the $+4$, or guessing $x=5$.

Question 3

1. $3.5$
2. $-3.5$
3. $6$ (correct answer)
4. $-6$

Explanation: To solve $7x - 5 = 3x + 19$, we need to isolate $x$ by moving all variable terms to one side and constants to the other. Subtract $3x$ from both sides: $7x - 3x - 5 = 19$, which gives us $4x - 5 = 19$. Add 5 to both sides: $4x = 24$. Divide by 4: $x = 6$. The common mistake is subtracting $3x$ from the wrong side or making sign errors when moving constants. Always perform the same operation on both sides of the equation.

Question 4

1. $80$ (correct answer)
2. $60$
3. $120$
4. $40$

Explanation: We need to solve $12 + 0.25x = 0.10x + 24$ for the number of texts $x$. First, let's move all $x$ terms to one side by subtracting $0.10x$ from both sides: $12 + 0.25x - 0.10x = 24$, which gives us $12 + 0.15x = 24$. Subtract 12 from both sides: $0.15x = 12$. Divide by 0.15: $x = 12 ÷ 0.15 = 80$. The key error to avoid is making arithmetic mistakes when subtracting decimal coefficients. Double-check decimal operations to ensure accuracy.

Question 5

1. $10$
2. $30$ (correct answer)
3. $23$
4. $3$

Explanation: We need to solve $0.5x + 7 = 0.2x + 16$ to find when the two phone plans cost the same. First, we collect like terms by subtracting $0.2x$ from both sides: $0.5x - 0.2x + 7 = 16$, which simplifies to $0.3x + 7 = 16$. Next, subtract 7 from both sides: $0.3x = 9$. Finally, divide by 0.3: $x = 9 ÷ 0.3 = 30$. The plans are equal at 30 gigabytes. A common mistake is incorrectly combining the decimal coefficients. To avoid decimal division errors, you can multiply the entire equation by 10 first to work with whole numbers.

Question 6

1. 8 (correct answer)
2. 3
3. -8
4. 4

Explanation: Add 5 to get $\frac{3}{4}x = 6$, then multiply by $\frac{4}{3}$ to find $x = 8$. The distractors come from adding or scaling incorrectly.

Question 7

1. $7$ (correct answer)
2. $\dfrac{7}{49}$
3. $\dfrac{49}{7}$
4. $-7$

Explanation: This problem asks us to find when two gym cost plans are equal by solving the equation $45 + 19x = 12x + 94$. To solve, we first collect like terms by subtracting 12x from both sides: $45 + 19x - 12x = 94$, which gives us $45 + 7x = 94$. Next, we subtract 45 from both sides: $7x = 49$. Finally, we divide both sides by 7: $x = 7$. A common error is to incorrectly combine the x terms or make arithmetic mistakes when moving constants. When solving equations with variables on both sides, always move all variable terms to one side first.

Question 8

1. $x=\frac{4}{3}(y+5)$ (correct answer)
2. $x=\frac{3}{4}(y+5)$
3. $x=\frac{4}{3}(y-5)$
4. $x=\frac{3}{4}(y-5)$

Explanation: We need to solve $y=\frac{3}{4}x-5$ for $x$ in terms of $y$. First, add 5 to both sides: $y + 5 = \frac{3}{4}x$. To isolate $x$, multiply both sides by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$: $\frac{4}{3}(y + 5) = \frac{4}{3} \cdot \frac{3}{4}x = x$. Therefore, $x = \frac{4}{3}(y + 5)$. A common mistake is incorrectly finding the reciprocal or forgetting to apply the operation to the entire expression $(y + 5)$. When solving for a variable, always perform the same operation on both sides of the equation.

Question 9

1. $19$
2. $12$ (correct answer)
3. $-12$
4. $-19$

Explanation: We need to solve $4x+7=2x+31$ with variables on both sides. Subtract $2x$ from both sides: $2x+7=31$. Subtract 7 from both sides: $2x=24$. Divide by 2: $x=12$. The key is to move all variable terms to one side and constants to the other by performing the same operation on both sides. Check your answer by substituting back: $4(12)+7=48+7=55$ and $2(12)+31=24+31=55$ ✓.

Question 10

1. $6$
2. $\frac{22}{3}$ (correct answer)
3. $\frac{14}{3}$
4. $\frac{20}{3}$

Explanation: We need to solve $\frac{3x}{4}-\frac{1}{2}=5$. First, find a common denominator of 4: $\frac{3x}{4}-\frac{2}{4}=5$. Combine: $\frac{3x-2}{4}=5$. Multiply both sides by 4: $3x-2=20$. Add 2: $3x=22$. Divide by 3: $x=\frac{22}{3}$. The common error is forgetting to multiply all terms by the same number when clearing fractions. Always check that you've multiplied every term, including constants.

Question 11

1. 2
2. 14
3. -3
4. 7 (correct answer)

Explanation: Distribute and collect terms: $5x - 10 = 3x + 4 \Rightarrow 2x = 14 \Rightarrow x = 7$. The other choices reflect forgetting to divide by 2 (14), moving terms incorrectly (-3), or ignoring the constant 10 (2).

Question 12

1. 8
2. 13
3. 5
4. 3 (correct answer)

Explanation: This problem asks us to solve the linear equation $3x + 5 = 2x + 8$ for the value of $x$. First, subtract $2x$ from both sides to get $3x - 2x + 5 = 8$, which simplifies to $x + 5 = 8$. Next, subtract $5$ from both sides to isolate $x$: $x = 8 - 5 = 3$. A common error would be incorrectly combining like terms or making arithmetic mistakes when subtracting. To verify, substitute $x = 3$ back into the original equation: $3(3) + 5 = 9 + 5 = 14$ and $2(3) + 8 = 6 + 8 = 14$, confirming our solution is correct.

Question 13

1. 80
2. 70
3. 65
4. 75 (correct answer)

Explanation: Solve $0.10t + 5 = 12.50$ to get $0.10t = 7.50$ and $t = 75$.

Question 14

1. 6
2. 7
3. 9
4. 8 (correct answer)

Explanation: Subtract 7 to get $4x = 32$, then divide by 4 to find $x = 8$.

Question 15

1. $r=\frac{IP}{t}$
2. $r=\frac{I}{Pt}$ (correct answer)
3. $r=\frac{Pt}{I}$
4. $r=\frac{It}{P}$

Explanation: We need to solve $I = Prt$ for $r$. To isolate $r$, we need to divide both sides by everything that's multiplied with $r$, which is $P$ and $t$. Dividing both sides by $Pt$: $\frac{I}{Pt} = \frac{Prt}{Pt} = r$. Therefore, $r = \frac{I}{Pt}$. The common mistake is dividing by only one of the factors (just $P$ or just $t$) or incorrectly placing variables in the numerator when they should be in the denominator. When solving for a variable, divide by all factors multiplied with that variable.

Question 16

1. $50$ (correct answer)
2. $14$
3. $32$
4. $\frac{82}{5}$

Explanation: We need to solve \frac{5}{9}(x - 32) \= 10 by isolating x. First, multiply both sides by $\frac{9}{5}$ to clear the fraction: x - 32 \= 10 \times \frac{9}{5} \= \frac{90}{5} \= 18. Then add 32 to both sides to get x \= 18 + 32 \= 50. A common mistake is to multiply by $\frac{5}{9}$ instead of its reciprocal $\frac{9}{5}$, which would give x - 32 \= 10 \times \frac{5}{9} \= \frac{50}{9} \approx 5.56, leading to $x \approx 37.56$. When a fraction multiplies a parenthetical expression, divide both sides by that fraction (multiply by its reciprocal) to isolate the parentheses.

Question 17

1. 7.2
2. 10.4 (correct answer)
3. 8.8
4. 9.6

Explanation: Write $3x + 2(x - 4) = 44 \Rightarrow 5x - 8 = 44$, so $x = 52/5 = 10.4$. Using $x$ for both ticket types (8.8), treating children as $x + 4$ (7.2), or misdistributing the 2 (9.6) are common errors.

Question 18

1. $12$
2. $13$ (correct answer)
3. $19$
4. $16$

Explanation: This problem asks us to find the monthly fee $x$ when the total cost after 6 months is $96, given by the equation $18 + 6x = 96$. To solve, we first subtract 18 from both sides: $6x = 96 - 18 = 78$. Then we divide both sides by 6: $x = 78 ÷ 6 = 13$. The key error to avoid is dividing 96 by 6 before subtracting the registration fee of $18. When solving equations with multiple terms, always isolate the variable term by moving constants to the other side first.

Question 19

1. 6
2. 10
3. 14
4. 18 (correct answer)

Explanation: Subtract 8 to get $\frac{x}{3} = 6$, then multiply by 3 to find $x = 18$. The other answers come from stopping at 6 or mixing the operations when isolating $x$.

Question 20

1. $500$ (correct answer)
2. $-500$
3. $1500$
4. $-1500$

Explanation: This problem asks when two phone plans cost the same by solving 35 + 0.08m = 50 + 0.05m. First, we collect like terms by subtracting 0.05m from both sides: 35 + 0.08m - 0.05m = 50, which gives us 35 + 0.03m = 50. Subtracting 35 from both sides: 0.03m = 15. Dividing by 0.03: m = 15/0.03 = 500. The key error to avoid is mishandling the decimal coefficients when combining like terms. When comparing plans, the intersection point represents where both options have equal value.