7th Grade Math Quiz: Solve Two Step Linear Equations

What this quiz covers

This quiz focuses on Solve Two Step Linear Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for 7th Grade 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. Width = 10 inches, Length = 32 inches
2. Width = 12 inches, Length = 30 inches (correct answer)
3. Width = 14 inches, Length = 28 inches
4. Width = 16 inches, Length = 26 inches

Explanation: Let $w$ be the width. Then length = $2w + 6$. Perimeter formula: $2w + 2(2w + 6) = 84$. Simplifying: $2w + 4w + 12 = 84$, so $6w = 72$ and $w = 12$. Therefore length = $2(12) + 6 = 30$. Check: $2(12) + 2(30) = 24 + 60 = 84$ ✓. Choice A results from solving $2w + 2w + 6 = 84$ (forgetting to double the length). Choice C uses incorrect coefficient setup. Choice D comes from arithmetic errors in solving the equation.

Question 2

1. Left side: $4(2 \cdot 13 - 3) = 4(20) = 80$; Right side: $6 \cdot 13 + 14 = 92$
2. Left side: $4(2 \cdot 13 - 3) = 4(23) = 92$; Right side: $6 \cdot 13 + 14 = 86$
3. Left side: $4(2 \cdot 13 - 3) = 4(26) = 104$; Right side: $6 \cdot 13 + 14 = 92$
4. Left side: $4(2 \cdot 13 - 3) = 4(23) = 92$; Right side: $6 \cdot 13 + 14 = 92$ (correct answer)

Explanation: When you have an algebraic equation and need to verify a solution, you substitute the proposed value back into the original equation and check that both sides are equal. This confirmation step is crucial because it catches any algebraic errors you might have made while solving. To verify $x = 13$ in the equation $4(2x - 3) = 6x + 14$, you need to carefully substitute 13 for $x$ and evaluate each side separately. For the left side: $4(2x - 3) = 4(2 \cdot 13 - 3) = 4(26 - 3) = 4(23) = 92$ For the right side: $6x + 14 = 6 \cdot 13 + 14 = 78 + 14 = 92$ Since both sides equal 92, the solution $x = 13$ is correct. This matches answer choice D. Looking at the wrong answers: Choice A makes an order of operations error on the left side, calculating $2 \cdot 13 - 3$ as 20 instead of 23, leading to 80 instead of 92. Choice B correctly calculates the left side as 92 but makes an arithmetic error on the right side, getting 86 instead of 92. Choice C makes a different error on the left side, somehow getting 26 instead of 23 inside the parentheses. Remember: verification problems test your arithmetic precision more than your algebra skills. Work slowly through the order of operations, double-check your basic arithmetic, and make sure both sides of the equation give you the same final value.

Question 3

1. $6$ tickets (correct answer)
2. $5$ tickets
3. $7$ tickets
4. $54$ tickets

Explanation: This question tests solving two-step equations from word problems in the $px + q = r$ form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve $px + q = r$, first subtract $q$ from both sides to isolate the variable term, for example, $9t + 4.5 = 58.5$ becomes $9t = 54$, then divide both sides by $p$ to isolate the variable, so $9t / 9 = 54 / 9$ gives $t = 6$; always verify by substituting back, like $9 \times 6 + 4.5 = 54 + 4.5 = 58.5$, which checks out. For this specific problem, set up the equation from the context: $9t + 4.5 = 58.5$, subtract $4.5$ to get $9t = 54$, divide by $9$ to find $t = 6$, verify $9 \times 6 + 4.5 = 58.5$, and interpret as $6$ tickets bought. The correct two-step process yields $t = 6$, which is choice A. Common errors include dividing first ($58.5 / 9 = 6.5$, subtract $4.5 = 2$ nonsense), arithmetic mistakes ($54 / 9 = 5$ wrongly), one-step only ($58.5 - 4.5 = 54$, forget divide), sign error (adding $4.5$ instead), or setup wrong ($4.5t + 9 = 58.5$). Strategy: identify the equation form $px + q = r$ from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense ($6$ tickets at $9$ is $54$ plus $4.50$ fee totals $58.50$), and check reasonableness (total near $6 \times 9 +$ small fee). Comparing to arithmetic, work backwards: $58.5 - 4.5 = 54$, $54 / 9 = 6$, same answer; algebra generalizes for any total; avoid mistakes like wrong order or stopping after one step.

Question 4

1. $3.50 + 2.25x = 25.75$; the trip was 9.9 miles total
2. $3.50 + 2.25(x - 1) = 25.75$; the trip was 10.9 miles total
3. $3.50 + 2.25(x - 1) = 25.75$; the trip was 11 miles total (correct answer)
4. $3.50x + 2.25(x - 1) = 25.75$; the trip was 6.2 miles total

Explanation: Since the first mile costs $3.50 and additional miles cost$2.25 each, if the total trip is $x$ miles, there are $(x-1)$ additional miles. The correct equation is $3.50 + 2.25(x-1) = 25.75$. Solving: $2.25(x-1) = 22.25$, so $x-1 = 22.25 ÷ 2.25 = 9.89$. Since distance must be a whole number of miles, $x-1 = 10$, giving $x = 11$ miles. Choice A uses the wrong equation structure. Choice B has the right equation but wrong calculation. Choice D incorrectly charges $3.50 per mile for all miles.

Question 5

1. 8 people, with both options costing $96 total
2. 9 people, with both options costing $108 total (correct answer)
3. 10 people, with both options costing $120 total
4. 11 people, with both options costing $132 total

Explanation: Let $n$ be the number of people. Option A costs $12n$ and Option B costs $36 + 8n$. Setting them equal: $12n = 36 + 8n$, so $4n = 36$ and $n = 9$. Check: Option A = $12(9) = 108$ and Option B = $36 + 8(9) = 36 + 72 = 108$. Both options cost exactly $108 for 9 people. The other choices show incorrect group sizes that don't result in equal costs.

Question 6

1. 8 hours
2. 5 hours
3. 6 hours (correct answer)
4. 7 hours

Explanation: This problem tests solving two-step equations from word problems in the form $px + q = r$, where we multiply the variable then add a constant, using inverse operations. The bike rental costs $12 per hour (variable cost) plus a$6 helmet fee (fixed cost), totaling $78, so we write$12h + 6 = 78$where h is the number of hours. To solve: (1) subtract 6 from both sides to isolate the variable term ($12h = 72$), then (2) divide by 12 to isolate h ($h = 72 ÷ 12 = 6$). The rental lasted 6 hours. Common errors include: stopping after one step (getting 72 as the answer), reversing operations (dividing first:$78 ÷ 12 = 6.5$, then subtracting 6 =$0.5$), or arithmetic mistakes ($72 ÷ 12 = 7$). Strategy: identify the equation form from context (hourly rate × hours + helmet fee = total), apply inverse operations in reverse order (subtract then divide), and verify by substituting back ($12 × 6 + 6 = 72 + 6 = 78 ✓$).

Question 7

1. $8$ months
2. $7$ months (correct answer)
3. $56$ months
4. $6$ months

Explanation: This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 8m + 3 = 59 becomes 8m = 56, then divide both sides by p to isolate the variable, so 8m / 8 = 56 / 8 gives m = 7; always verify by substituting back, like 8 × 7 + 3 = 56 + 3 = 59, which checks out. For this specific problem, set up the equation from the context: 8m + 3 = 59, subtract 3 to get 8m = 56, divide by 8 to find m = 7, verify 8 × 7 + 3 = 59, and interpret as 7 months of subscription. The correct two-step process yields m = 7, which is choice A. Common errors include one-step only (59 - 3 = 56, forgetting to divide so m = 56), operation order reversed (59 / 8 = 7.375, then subtract 3 ≈ 4.375 nonsense), sign error (adding 3 instead: 8m = 62), arithmetic mistake (56 / 8 = 8 wrongly), or setup wrong (3m + 8 = 59 reversing). Strategy: identify the equation form px + q = r from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense (7 months at $8 is$56 plus $3 fee totals$59), and check reasonableness (total around 7 × 8 + small fee). Comparing to arithmetic, work backwards: 59 - 3 = 56, 56 / 8 = 7, same answer; algebra generalizes for any total; avoid mistakes like stopping after one step or wrong order.

Question 8

1. $x=70$
2. $x=8$ (correct answer)
3. $x=9$
4. $x=6$

Explanation: This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 8.75x + 10 = 80 becomes 8.75x = 70, then divide both sides by p to isolate the variable, so 8.75x ÷ 8.75 = 70 ÷ 8.75 gives x = 8; for the other form p(x + q) = r, divide by p first then subtract q, and always verify by substituting back, like 8.75 × 8 + 10 = 70 + 10 = 80, which checks out in the context of cookie dough tubs at $8.75 each plus$10 fee totaling $80 for x tubs. For this specific problem, the equation is given as 8.75x + 10 = 80, subtract 10 to get 8.75x = 70, divide by 8.75 to find x = 8, verify 8.75 × 8 + 10 = 80, and interpret as 8 tubs bought. The correct two-step process yields x = 8, which is choice A. Common errors include subtracting only and forgetting to divide (x = 70, choice C), reversing order by dividing first (80 ÷ 8.75 ≈ 9.14 then subtract 10, nonsensical), sign errors like adding instead (8.75x = 90), arithmetic mistakes (70 ÷ 8.75 = 9 or 6), or wrong setup though it's provided. Strategy tips include identifying the px + q = r form from per-tub cost plus fixed fee, applying inverse operations in reverse (subtract then divide), maintaining equality by doing the same to both sides, verifying by substitution and context sense (8 tubs at$8.75 is $70 plus$10 totals $80, reasonable), and checking reasonableness. Compared to arithmetic working backwards (80 - 10 = 70, 70 ÷ 8.75 = 8), algebra generalizes for any total; avoid mistakes like stopping after one step, wrong order, sign errors, or mismatched setup.

Question 9

1. h=4 (correct answer)
2. h=7/2
3. h=3
4. h=13/3

Explanation: Subtracting 6 from both sides of 18h + 6 = 78 gives 18h = 72, and dividing by 18 gives h = 4, matching choice A. Checking: 18 x 4 + 6 = 72 + 6 = 78, which fits the situation of 4 hours tutored. Choices B, C, and D don't satisfy the equation: substituting each back in gives totals of $69,$60, and $84 instead of the required$78.

Question 10

1. 7 months
2. 6 months
3. 4 months
4. 5 months (correct answer)

Explanation: This problem tests solving two-step equations from word problems in the form $px + q = r$, where we multiply the variable then add a constant, using inverse operations. The streaming service costs $9.75 per month (variable cost) plus a$4.50 fee (fixed cost), totaling $53.25, so we write$9.75m + 4.50 = 53.25$where m is the number of months. To solve algebraically: (1) subtract 4.50 from both sides ($9.75m = 48.75$), then (2) divide by 9.75 ($m = 48.75 ÷ 9.75 = 5$). Working backward gives the same result:$$53.25 - $4.50 = $48.75$, then$$48.75 ÷ $9.75 = 5$ months. Common errors include: stopping after one step, reversing operations, or arithmetic mistakes with decimals. Both methods—algebraic equation solving and working backward—yield 5 months, demonstrating that algebra generalizes the arithmetic approach of undoing operations in reverse order.

Question 11

1. $x=16$
2. $x=42$
3. $x=12$
4. $x=14$ (correct answer)

Explanation: This question tests solving two-step equations from word problems in the px + q = r form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve px + q = r, first subtract q from both sides to isolate the variable term, for example, 3x + 5 = 47 becomes 3x = 42, then divide both sides by p to isolate the variable, so 3x / 3 = 42 / 3 gives x = 14; always verify by substituting back, like 3 × 14 + 5 = 42 + 5 = 47, which checks out. For this specific problem, use the given model 3x + 5 = 47, subtract 5 to get 3x = 42, divide by 3 to find x = 14, verify 3 × 14 + 5 = 47, and interpret as 14 gift bags can be made. The correct two-step process yields x = 14, which is choice A. Common errors include dividing first (47 / 3 ≈ 15.667, subtract 5 ≈ 10.667 nonsense), arithmetic mistakes (42 / 3 = 12 or 16 wrongly), one-step only (47 - 5 = 42, forget divide), sign error (3x = 52 by adding), or setup wrong (5x + 3 = 47). Strategy: identify the equation form px + q = r from the context, apply inverse operations in reverse order (subtract then divide to undo add then multiply), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation and makes sense (14 bags need 42 stickers plus 5 extra totals 47), and check reasonableness (total minus extra divisible by 3). Comparing to arithmetic, work backwards: 47 - 5 = 42, 42 / 3 = 14, same answer; algebra generalizes for any total; avoid mistakes like wrong order or miscalculation.

Question 12

1. $x=5$ (correct answer)
2. $x=13$
3. $x=7$
4. $x=3$

Explanation: This question tests solving two-step equations in the p(x + q) = r form, where you multiply a grouped expression by a coefficient, using inverse operations to isolate the variable. To solve p(x + q) = r, first divide both sides by p to isolate the grouped term, for example, 3(x + 4) = 27 becomes x + 4 = 9, then subtract q from both sides to isolate the variable, so x = 9 - 4 = 5; always verify by substituting back, like 3(5 + 4) = 3 × 9 = 27, which checks out. For this specific problem, start with 3(x + 4) = 27, divide by 3 to get x + 4 = 9, subtract 4 to find x = 5, and verify 3(5 + 4) = 27. The correct two-step process yields x = 5, which is choice A. Common errors include distributing wrongly (3x + 4 = 27, then mistakes), order reversed (subtract 4 from 27 = 23, divide by 3 ≈ 7.67 nonsense), arithmetic error (27 / 3 = 8, 8 - 4 = 4), or expanding incorrectly (3x + 12 = 27). Strategy: identify the equation form p(x + q) = r, apply inverse operations in reverse order (divide then subtract to undo multiply after add), maintain equality by doing the same to both sides, verify by substituting back to ensure it satisfies the equation, and check reasonableness (5 + 4 = 9, times 3 is 27). Comparing to arithmetic, work backwards: 27 / 3 = 9, 9 - 4 = 5, same answer; algebra generalizes for any right side; avoid mistakes like wrong order or distribution errors.

Question 13

1. $b=8$ (correct answer)
2. $b=26$
3. $b=7$
4. $b=9$

Explanation: This question tests solving two-step equations from word problems in the $px + q = r$ form, where you multiply a variable by a coefficient and then add a constant, using inverse operations to isolate the variable. To solve $px + q = r$, first subtract $q$ from both sides to isolate the variable term, for example, $3.25b + 1.5 = 27.5$ becomes $3.25b = 26$, then divide both sides by $p$ to isolate the variable, so $3.25b \div 3.25 = 26 \div 3.25$ gives $b = 8$; for the other form $p(x + q) = r$, divide by $p$ first then subtract $q$, and always verify by substituting back, like $3.25 \times 8 + 1.5 = 26 + 1.5 = 27.5$, which checks out in the context of water bottles at $3.25$ each plus $1.50$ tax totaling $27.50$ for $b$ bottles. For this specific problem, the equation is given as $3.25b + 1.5 = 27.5$, subtract $1.5$ to get $3.25b = 26$, divide by $3.25$ to find $b = 8$, verify $3.25 \times 8 + 1.5 = 27.5$, and interpret as $8$ water bottles bought. The correct two-step process yields $b = 8$, which is choice A. Common errors include subtracting only and forgetting to divide ($b = 26$, choice D), reversing order by dividing first ($27.5 \div 3.25 \approx 8.46$ then subtract $1.5$, nonsensical), sign errors like adding instead ($3.25b = 29$), arithmetic mistakes ($26 \div 3.25 = 9$ or $7$), or wrong setup though it's provided. Strategy tips include identifying the $px + q = r$ form from per-bottle cost plus fixed tax, applying inverse operations in reverse (subtract then divide), maintaining equality by doing the same to both sides, verifying by substitution and context sense ($8$ bottles at $3.25$ is $26$ plus $1.50$ totals $27.50$, reasonable), and checking reasonableness. Compared to arithmetic working backwards ($27.5 - 1.5 = 26$, $26 \div 3.25 = 8$), algebra generalizes for any total; avoid mistakes like stopping after one step, wrong order, sign errors, or mismatched setup.

Question 14

1. m=6.6
2. m=5
3. m=6 (correct answer)
4. m=7

Explanation: The equation is 7.5m + 4.5 = 49.5. Subtracting 4.5 from both sides gives 7.5m = 45, and dividing by 7.5 gives m = 6, matching choice C. Checking: 7.5 x 6 + 4.5 = 45 + 4.5 = 49.5, which fits the context of 6 months of subscription. Choice A (6.6) comes from dividing 49.5 by 7.5 first instead of subtracting the fee, and choices B and D don't satisfy the equation when substituted back in.

Question 15

1. $p=9$
2. $p=7$ (correct answer)
3. $p=11$
4. $p=5$

Explanation: To solve 5(p+2)=45, first divide both sides by 5: p+2=9. Then subtract 2 from both sides: p=7, matching choice B. Verify: 5(7+2)=5x9=45, which checks out. Choice A stops after dividing but forgets to subtract 2, leaving p+2's value (9) as the final answer. Choice C divides 45 by 5 to get 9, then adds 2 instead of subtracting, giving 11. Choice D makes an arithmetic error, subtracting incorrectly to arrive at 5.

Question 16

1. $6x-3=51$, so $x=9$
2. $3x+6=51$, so $x=15$
3. $51x+3=6$, so $x=\tfrac{1}{17}$
4. $6x+3=51$, so $x=8$ (correct answer)

Explanation: This problem tests solving two-step equations from word problems in the form px+q=r, where we multiply the variable then add a constant, using inverse operations. The class makes 6 bags with x pretzels each (6x total) plus 3 leftover pretzels equals 51 total pretzels, giving equation 6x + 3 = 51. To solve: (1) subtract 3 from both sides to isolate the variable term (6x = 48), then (2) divide by 6 to isolate x (x = 48 ÷ 6 = 8). Each bag has 8 pretzels, matching answer choice A. Common errors include: reversing the equation setup (51x + 3 = 6 as in choice C), swapping coefficients (3x + 6 = 51 as in choice D), or using subtraction instead of addition for leftovers (6x - 3 = 51 as in choice B). Strategy: identify what's multiplied (bags × pretzels per bag) and what's added (leftovers), set up equation matching the context, apply inverse operations, and verify (6×8+3=48+3=51✓).

Question 17

1. 6.25 combos
2. 6 combos (correct answer)
3. 5 combos
4. 8 combos

Explanation: The equation for this situation is 6.25x + 2.5 = 40, where x is the number of combos. Subtracting 2.5 from both sides gives 6.25x = 37.5, and dividing both sides by 6.25 gives x = 6. Checking: 6.25(6) + 2.5 = 37.5 + 2.5 = 40, which matches. Choice A mistakes the price per combo for the number of combos. Choice C comes from an arithmetic slip when dividing 37.5 by 6.25. Choice D comes from dividing the total by the price per combo first, 40/6.25 = 6.4, without properly subtracting the fee first.

Question 18

1. $7$ friends (correct answer)
2. $9$ friends
3. $11$ friends
4. $5$ friends

Explanation: Since you're buying one item for each friend plus 2 extra items for yourself and a backup, the total number of items is x + 2, where x is the number of friends. The equation for the total cost is 4(x + 2) = 36. Dividing both sides by 4 gives x + 2 = 9, and subtracting 2 gives x = 7. Checking: 4(7 + 2) = 4(9) = 36, which matches. Choice B comes from stopping after finding the total number of items, 9, without subtracting the 2 extra. Choice C and choice D come from errors in the order of the division and subtraction steps.

Question 19

1. 12 packs
2. 10 packs
3. 9 packs
4. 8 packs (correct answer)

Explanation: The situation can be written as the equation 3.25p + 5.50 = 31.50, where p is the number of packs. Subtracting 5.50 from both sides gives 3.25p = 26.00, and dividing both sides by 3.25, the cost per pack, gives p = 8, matching Choice D. Choice A, 12 packs, and Choice B, 10 packs, do not satisfy the equation when checked. Choice C, 9 packs, is close but does not check out either, since 3.25 x 9 + 5.50 = 34.75, not 31.50. Checking the answer, 3.25 x 8 + 5.50 = 26.00 + 5.50 = 31.50, which confirms 8 packs is correct.

Question 20

1. $x=7$
2. $x=42$
3. $x=5$
4. $x=6$ (correct answer)

Explanation: Starting with the equation 7x plus 2 equals 44, subtracting 2 from both sides gives 7x equals 42, and dividing both sides by 7 gives x equals 6, matching choice D. Choice A, x equals 7, does not satisfy the equation, since 7 times 7 plus 2 equals 51, not 44. Choice B, x equals 42, comes from stopping after subtracting 2 and forgetting to divide by 7. Choice C, x equals 5, comes from an arithmetic slip when dividing 42 by 7. Checking the answer by substitution, 7 times 6 plus 2 equals 42 plus 2, which equals 44, confirms that x equals 6 is correct.