Multi-Step Equations

Grade 7 · algebra · 70 practice problems · read aloud

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Multi-Step Equations

What Are Multi-Step Equations? 🤔

Multi-step equations are algebraic equations that require more than one operation to solve for the variable. You'll need to use a combination of addition, subtraction, multiplication, and division to find the value of the unknown (like x). Mastering this is crucial because it's the foundation for solving almost all future algebra problems!

How to Solve: Your Step-by-Step Guide

  1. Simplify Both Sides: Use the Distributive Property and combine any like terms on each side of the equals sign.
  2. Move Variable Terms: Use addition or subtraction to get all the variable terms on one side of the equation and the constants on the other.
  3. Isolate the Variable: Use multiplication or division to get the variable by itself.
  4. Check Your Solution: Always plug your answer back into the original equation to make sure it works!

Worked Examples

Example 1: Solve for x: 3x + 5 = 14

  1. Subtract 5 from both sides: 3x + 5 - 5 = 14 - 5 → 3x = 9
  2. Divide both sides by 3: 3x / 3 = 9 / 3
  3. Solution: x = 3

Example 2: Solve for n: 2(n - 4) = 10

  1. Distribute the 2: 2n - 8 = 10
  2. Add 8 to both sides: 2n - 8 + 8 = 10 + 8 → 2n = 18
  3. Divide both sides by 2: 2n / 2 = 18 / 2
  4. Solution: n = 9

Common Mistakes to Avoid 🚫

  • Sign Errors: Forgetting to change the sign when moving a term across the equals sign. If you subtract from one side, you MUST subtract from the other.
  • Distributive Property Errors: Forgetting to multiply the number outside the parenthesis by every term inside. Example: 2(x + 3) = 2x + 6, not 2x + 3.
  • Order of Operations: Not combining like terms before moving variables.

Tips & Tricks

  • Think of the equation like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced!
  • SADMEP: This is the reverse of PEMDAS. When solving, you often undo operations in the reverse order (Subtraction/Addition first, then Division/Multiplication, then Exponents/Parentheses).
  • Circle or box your final answer so it's clear.

How to Practice

Start with simple two-step equations and gradually try problems with more steps or that require using the distributive property. You can find practice worksheets online, use flashcards with different problems on the front and answers on the back, or teach the steps to a friend—teaching is a great way to learn!

Practice problems

6 of the 70, worked through step by step — try them before opening the answer.

1 3(x + 5) - 7 = 2x + 14

Hint: First distribute any multiplication across terms in parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 6

  1. Distribute the 3** Multiply 3 by both terms inside the parentheses: 3 * x + 3 * 5 - 7 = 2x + 14 This gives: 3x + 15 - 7 = 2x + 14 **
  2. Simplify the left side** Combine 15 - 7: 3x + 8 = 2x + 14 **
  3. Move x terms to one side** Subtract 2x from both sides: 3x - 2x + 8 = 14 x + 8 = 14 **
  4. Isolate x** Subtract 8 from both sides: x + 8 - 8 = 14 - 8 x = 6 **
  5. Check the solution** Plug x = 6 into the original equation: 3(6 + 5) - 7 = 2(6) + 14 3(11) - 7 = 12 + 14 33 - 7 = 26 26 = 26 ✓ **Final answer:** x = 6

Let's solve the equation step-by-step. We start with: 3(x + 5) - 7 = 2x + 14 **

2 3(2x - 5) + 7 = 4x + 13

Hint: First distribute any multiplication across terms in parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 10.5

  1. Distribute the 3** Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 So the equation becomes: 6x - 15 + 7 = 4x + 13 --- **
  2. Combine like terms on the left** -15 + 7 = -8 So now: 6x - 8 = 4x + 13 --- **
  3. Move x terms to one side** Subtract 4x from both sides: 6x - 4x - 8 = 4x - 4x + 13 2x - 8 = 13 --- **
  4. Move constant terms to the other side** Add 8 to both sides: 2x - 8 + 8 = 13 + 8 2x = 21 --- **
  5. Solve for x** Divide both sides by 2: x = 21/2 x = 10.5 --- **Final answer:** x = 10.5

Let's solve the equation step-by-step. We start with: 3(2x - 5) + 7 = 4x + 13 --- **

3 3(x + 5) - 2(2x - 1) = 4

Hint: Begin by distributing any coefficients outside parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: 13

  1. Distribute the numbers outside the parentheses** 3(x + 5) becomes 3*x + 3*5 = 3x + 15 -2(2x - 1) becomes -2*2x -2*(-1) = -4x + 2 So the equation becomes: 3x + 15 - 4x + 2 = 4 **
  2. Combine like terms on the left side** 3x - 4x = -1x (or -x) 15 + 2 = 17 So now we have: -x + 17 = 4 **
  3. Isolate the term with x** Subtract 17 from both sides: -x + 17 - 17 = 4 - 17 -x = -13 **
  4. Solve for x** Multiply both sides by -1: x = 13 **Final check:** Substitute x = 13 into the original equation: 3(13 + 5) - 2(2*13 - 1) = 3(18) - 2(26 - 1) = 54 - 2(25) = 54 - 50 = 4 This matches the right-hand side, so the solution is correct. **Answer:** x = 13

Let's solve the equation step by step. We start with: 3(x + 5) - 2(2x - 1) = 4 **

4 3(x - 4) + 7 = 2x + 5 = ?

Hint: First distribute any multiplication across parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: 10

  1. Expand the left side** 3(x - 4) means 3 times x and 3 times -4: 3*x - 3*4 + 7 = 2x + 5 This simplifies to: 3x - 12 + 7 = 2x + 5 **
  2. Combine like terms on the left** -12 + 7 = -5 So we have: 3x - 5 = 2x + 5 **
  3. Get all x terms on one side** Subtract 2x from both sides: 3x - 2x - 5 = 2x - 2x + 5 x - 5 = 5 **
  4. Isolate x** Add 5 to both sides: x - 5 + 5 = 5 + 5 x = 10 **
  5. Check the solution** Substitute x = 10 into the original equation: 3(10 - 4) + 7 = 3(6) + 7 = 18 + 7 = 25 2(10) + 5 = 20 + 5 = 25 Both sides are equal, so the solution is correct. **Final answer:** x = 10

Let's solve the equation step by step. We start with: 3(x - 4) + 7 = 2x + 5 **

5 3(2x - 7) + 4 = 5(x + 2) - 8

Hint: Start by distributing the numbers outside the parentheses to eliminate them, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 11

  1. Distribute the 3 on the left side: 3 * 2x = 6x and 3 * (-7) = -21, so left side becomes 6x - 21 + 4
  2. Distribute the 5 on the right side: 5 * x = 5x and 5 * 2 = 10, so right side becomes 5x + 10 - 8
  3. Combine like terms on left side: -21 + 4 = -17, so left side is 6x - 17
  4. Combine like terms on right side: 10 - 8 = 2, so right side is 5x + 2
  5. Equation is now 6x - 17 = 5x + 2
  6. Subtract 5x from both sides: 6x - 5x - 17 = 2, which gives x - 17 = 2
  7. Add 17 to both sides: x - 17 + 17 = 2 + 17, which gives x = 19
  8. Check: 3(2*19 - 7) + 4 = 3(38 - 7) + 4 = 3(31) + 4 = 93 + 4 = 97, and 5(19 + 2) - 8 = 5(21) - 8 = 105 - 8 = 97 Both sides equal 97, so x = 19 is correct.

6 3(2x - 7) + 5 = 4(x + 2) - 3

Hint: Start by distributing the numbers outside the parentheses to eliminate them, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 9

  1. Distribute on both sides 3(2x - 7) + 5 = 4(x + 2) - 3 6x - 21 + 5 = 4x + 8 - 3
  2. Combine like terms on each side 6x - 16 = 4x + 5
  3. Get all x terms on one side by subtracting 4x from both sides 6x - 16 - 4x = 4x + 5 - 4x 2x - 16 = 5
  4. Isolate the x term by adding 16 to both sides 2x - 16 + 16 = 5 + 16 2x = 21
  5. Solve for x by dividing both sides by 2 2x ÷ 2 = 21 ÷ 2 x = 10.5

The answer is x = 10.5.

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