Middle School Math Quiz: Two Step Equations

9 questions · exam conditions

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Two Step EquationsQuestion 1 of 9

The perimeter of a rectangle is 36 inches. If the width is 7 inches, which equation can be used to find the length $\ell$ ?

2\ell + 7 = 36

2(\ell + 7) = 72

\ell + 7 = 36

2\ell + 14 = 36

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Middle School Math Quiz

Middle School Math Quiz: Two Step Equations

Practice Two Step Equations in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Two Step Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School 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.

All questions

Question 1

The perimeter of a rectangle is 36 inches. If the width is 7 inches, which equation can be used to find the length $\ell$ ?

$2\ell + 7 = 36$
$2(\ell + 7) = 72$
$\ell + 7 = 36$
$2\ell + 14 = 36$ (correct answer)

Explanation: When you encounter rectangle perimeter problems, remember that perimeter equals the sum of all four sides. For a rectangle, this means adding length + width + length + width, which gives us the formula: Perimeter =

2\ell + 2w

. Given that the perimeter is 36 inches and the width is 7 inches, you can substitute these values into the formula:

36 = 2\ell + 2(7)

. Simplifying the width term:

36 = 2\ell + 14

. Rearranging this equation gives you

2\ell + 14 = 36

, which matches answer choice D. Let's examine why the other options are incorrect. Choice A (

2\ell + 7 = 36

) only accounts for one width instead of two, forgetting that a rectangle has two width sides. Choice B (

2(\ell + 7) = 72

) incorrectly uses 72 instead of the given perimeter of 36. Choice C (

\ell + 7 = 36

) completely ignores that perimeter involves all four sides, treating it as if you're just adding length and width once. To verify answer D is correct, you can solve it:

2\ell + 14 = 36

means

2\ell = 22

, so

\ell = 11

. Check:

2(11) + 2(7) = 22 + 14 = 36

✓ Study tip: Always write out the perimeter formula first (

P = 2\ell + 2w

), then substitute the known values. This systematic approach prevents you from accidentally forgetting that rectangles have two of each side.

Question 2

For what value of $k$ does the equation $\frac{2x - k}{5} = 4$ have solution $x = 9$ ?

$k = -2$ (correct answer)
$k = 2$
$k = 11$
$k = 38$

Explanation: Substitute

x = 9

into the equation:

\frac{2(9) - k}{5} = 4

, so

\frac{18 - k}{5} = 4

. Multiply by 5:

18 - k = 20

. Solve:

-k = 2

, so

k = -2

. Choice B gives

k = 2

from sign error. Choice C comes from

18 - 7 = 11

. Choice D comes from

18 + 20 = 38

Question 3

A cell phone plan charges a monthly fee of $25 plus $0.10 per text message. If someone's bill was $32.50 for one month, which equation represents the number of text messages $ $t$ $ they sent?

$0.10t - 25 = 32.50$
$25t + 0.10 = 32.50$
$25 + 0.10t = 32.50$ (correct answer)
$25(t + 0.10) = 32.50$

Explanation: When you encounter word problems about costs or pricing, your goal is to translate the real-world situation into a mathematical equation by identifying the fixed costs and variable costs. In this cell phone plan, there are two components: a fixed monthly fee of $25 (this stays the same regardless of usage) and a variable cost of $0.10 per text message (this depends on how many texts are sent). The total bill equals the fixed cost plus the variable cost, so: Total Bill = Fixed Fee + (Cost per text × Number of texts). Substituting the given values: $

32.50 = 25 + 0.10t

, which can be rewritten as

25 + 0.10t = 32.50

$. This matches answer choice C. Let's examine why the other options are incorrect. Choice A (

0.10t - 25 = 32.50

) subtracts the monthly fee instead of adding it, which doesn't make sense since fees increase your bill. Choice B (

25t + 0.10 = 32.50

) incorrectly multiplies the monthly fee by the number of texts and treats the per-text cost as a fixed amount. Choice D (

25(t + 0.10) = 32.50

) uses distribution incorrectly, suggesting the monthly fee is multiplied by both the number of texts and the per-text rate. Study tip: In cost problems, always identify what stays constant (fixed costs) and what changes based on usage (variable costs). Write the equation as: Total = Fixed + (Rate × Variable). This pattern appears frequently in pre-algebra word problems involving phone plans, taxi fares, and rental costs.

Question 4

Sarah needs to solve $7 - 2x = 21$ . She decides to first add $2x$ to both sides, then subtract 21 from both sides. What equation will she have after these two steps?

$28 = 2x$
$-14 = -2x$
$-14 = 2x$ (correct answer)
$7 = 21 + 2x$

Explanation: When solving equations, you need to carefully track what happens to both sides when you apply operations. This question tests whether you can follow a specific sequence of steps, even if they're not the most efficient path to the solution. Let's follow Sarah's steps exactly. Starting with

7 - 2x = 21

, she first adds

2x

to both sides:

Left side: $7 - 2x + 2x = 7$
Right side: $21 + 2x$
Result: $7 = 21 + 2x$

Next, she subtracts 21 from both sides:

Left side: $7 - 21 = -14$
Right side: $21 + 2x - 21 = 2x$
Final result: $-14 = 2x$

This confirms answer choice C is correct. Looking at the wrong answers: A)

28 = 2x

incorrectly adds 7 and 21 instead of subtracting. B)

-14 = -2x

makes the error of keeping the negative sign on the

x

term, forgetting that adding

2x

-2x

eliminates the negative. D)

7 = 21 + 2x

is actually the intermediate step after just the first operation, not the final result after both steps. When working through multi-step problems, always perform operations one at a time and double-check each step before moving on. Even if someone chooses an unusual order of operations (like Sarah did here), the algebra still works as long as you apply each step correctly to both sides.

Question 5

The equation $\frac{x + 5}{3} = 8$ has the same solution as which of the following equations?

$x + 5 = 11$
$x + 5 = 24$ (correct answer)
$\frac{x}{3} + 5 = 8$
$3x + 15 = 8$

Explanation: Solving

\frac{x + 5}{3} = 8

: multiply both sides by 3 to get

x + 5 = 24

, then

x = 19

. Choice B gives the same equation after the first step. Choice A would give

x = 6

, Choice C would give

x = 9

, and Choice D would give

x = -\frac{7}{3}

Question 6

Maria solved the equation $3x - 7 = 14$ and got $x = 3$ . When she substituted her answer back into the original equation to check, what value did she get for the left side of the equation?

$2$ (correct answer)
$7$
$14$
$21$

Explanation: When substituting

x = 3

into

3x - 7

, we get

3(3) - 7 = 9 - 7 = 2

. Since Maria's answer gives 2 on the left side but the equation should equal 14, her solution is incorrect. The correct solution is

x = 7

Question 7

Jake's work solving $4x + 9 = 1$ is shown below:

Step 1: $4x + 9 = 1$ Step 2: $4x = 1 + 9$ Step 3: $4x = 10$ Step 4: $x = \frac{10}{4} = 2.5$

In which step did Jake make his first error?

Step 1 (the original equation is incorrect)
Step 2 (incorrect operation with the constant term) (correct answer)
Step 3 (arithmetic error in addition)
Step 4 (incorrect division of fractions)

Explanation: In Step 2, Jake should have subtracted 9 from both sides to get

4x = 1 - 9 = -8

, not

4x = 1 + 9

. This is a common error when moving terms across the equals sign. The correct solution is

x = -2

. Steps 3 and 4 follow correctly from his error in Step 2.

Question 8

Which of the following equations is equivalent to $5 = 3y - 7$ ?

$5 - 7 = 3y$ and $y = -\frac{2}{3}$
$-2 = 3y$ and $y = -\frac{2}{3}$
$5 + 7 = 3y$ and $y = \frac{5}{3}$
$12 = 3y$ and $y = 4$ (correct answer)

Explanation: When solving linear equations, your goal is to isolate the variable by performing the same operation on both sides of the equation. This maintains the equation's balance while simplifying it step by step. Starting with

5 = 3y - 7

, you need to get

y

by itself. First, add 7 to both sides to eliminate the

-7

on the right side:

5 + 7 = 3y - 7 + 7

, which simplifies to

12 = 3y

. Next, divide both sides by 3 to isolate

y

\frac{12}{3} = \frac{3y}{3}

, giving you

y = 4

. You can verify this by substituting back:

5 = 3(4) - 7 = 12 - 7 = 5

✓ Choice D is correct because it shows both the intermediate step

12 = 3y

and the final answer

y = 4

. Choice A incorrectly subtracts 7 from the left side instead of adding it to both sides. This gives

5 - 7 = -2

, not the correct

12

. Choice B shows the wrong intermediate step

-2 = 3y

(from the error in choice A) but then correctly divides to get

y = -\frac{2}{3}

, which is wrong because the intermediate step was wrong. Choice C correctly adds 7 to get

5 + 7 = 3y

, but then makes an arithmetic error, claiming

5 + 7 = 5

instead of

12

, leading to the incorrect answer

y = \frac{5}{3}

. Remember: always perform the same operation on both sides of an equation, and double-check your arithmetic at each step to avoid simple calculation errors.

Question 9

If $2(n - 4) = 18$ , what is the value of $n - 4$ ?

$5$
$9$ (correct answer)
$13$
$22$

Explanation: From

2(n - 4) = 18

, divide both sides by 2 to get

n - 4 = 9

. We don't need to solve for

n

itself. Choice A comes from solving

n - 4 = 9

to get

n = 13

, then incorrectly using 5. Choice C gives the value of

n

, not

n - 4

. Choice D comes from adding instead of dividing.