Solving Algebra Equations: Grade 6 Step-by-Step Examples

What does it mean to solve an equation?

To solve an equation means to find the value of the variable that makes the equation true.

In x + 6 = 14, the value x = 8 works because 8 + 6 = 14.

In 4x = 24, the value x = 6 works because 4 * 6 = 24.

The balance idea

Think of an equation like a balanced scale.

The left side and right side must stay equal.

Whatever you do to one side, you must do to the other side.

That is why we subtract from both sides, add to both sides, divide both sides, or multiply both sides.

Inverse operations

Inverse operations undo each other.

Addition is undone by subtraction.

Subtraction is undone by addition.

Multiplication is undone by division.

Division is undone by multiplication.

The fastest way to solve a Grade 6 equation is to ask: "What operation is attached to the variable, and how do I undo it?"

Solving x + p = q

The form x + p = q means a number plus p equals q.

To solve it, subtract p from both sides.

Rule: if x + p = q, then x = q - p.

Example 1: x + 6 = 14

Problem: Solve x + 6 = 14.

Step 1: The variable has 6 added to it.

Step 2: Undo addition by subtracting 6 from both sides: x + 6 - 6 = 14 - 6.

Step 3: Simplify: x = 8.

Check: 8 + 6 = 14. True.

Answer: x = 8.

Example 2: x + 18 = 47

Problem: Solve x + 18 = 47.

Step 1: The variable has 18 added to it.

Step 2: Subtract 18 from both sides: x = 47 - 18.

Step 3: Calculate: 47 - 18 = 29.

Check: 29 + 18 = 47.

Answer: x = 29.

Example 3: x + 4.5 = 12

Problem: Solve x + 4.5 = 12.

Step 1: Undo the added 4.5.

Step 2: Subtract 4.5 from both sides: x = 12 - 4.5.

Step 3: Calculate: x = 7.5.

Check: 7.5 + 4.5 = 12.

Answer: x = 7.5.

Solving x - p = q

The form x - p = q means a number minus p equals q.

To solve it, add p to both sides.

Rule: if x - p = q, then x = q + p.

Example 4: x - 9 = 21

Problem: Solve x - 9 = 21.

Step 1: The variable has 9 subtracted from it.

Step 2: Undo subtraction by adding 9 to both sides: x - 9 + 9 = 21 + 9.

Step 3: Simplify: x = 30.

Check: 30 - 9 = 21.

Answer: x = 30.

Example 5: x - 3/4 = 2

Problem: Solve x - 3/4 = 2.

Step 1: Undo subtraction by adding 3/4 to both sides.

Step 2: x = 2 + 3/4.

Step 3: Write 2 as fourths: 2 = 8/4.

Step 4: Add: 8/4 + 3/4 = 11/4.

Answer: x = 11/4, or 2 3/4.

Solving px = q

The form px = q means p times x equals q.

To solve it, divide both sides by p.

Rule: if px = q, then x = q / p.

Example 6: 4x = 24

Problem: Solve 4x = 24.

Step 1: Read 4x as 4 * x.

Step 2: Undo multiplication by dividing both sides by 4: 4x / 4 = 24 / 4.

Step 3: Simplify: x = 6.

Check: 4 * 6 = 24.

Answer: x = 6.

Example 7: 7x = 63

Problem: Solve 7x = 63.

Step 1: The variable is multiplied by 7.

Step 2: Divide both sides by 7: x = 63 / 7.

Step 3: Calculate: x = 9.

Check: 7 * 9 = 63.

Answer: x = 9.

Example 8: 12 = 3x

Problem: Solve 12 = 3x.

Step 1: The variable side is on the right, but the equation still works the same way.

Step 2: Rewrite it as 3x = 12 if that feels easier.

Step 3: Divide both sides by 3: x = 12 / 3.

Step 4: Calculate: x = 4.

Answer: x = 4.

Example 9: 0.5x = 8

Problem: Solve 0.5x = 8.

Step 1: 0.5x means half of x.

Step 2: Divide both sides by 0.5: x = 8 / 0.5.

Step 3: Dividing by one half doubles the number: x = 16.

Check: 0.5 * 16 = 8.

Answer: x = 16.

Solving x / p = q

The form x / p = q means a number divided by p equals q.

To solve it, multiply both sides by p.

Rule: if x / p = q, then x = p * q.

Example 10: x / 5 = 9

Problem: Solve x / 5 = 9.

Step 1: The variable is divided by 5.

Step 2: Undo division by multiplying both sides by 5: x = 9 * 5.

Step 3: Calculate: x = 45.

Check: 45 / 5 = 9.

Answer: x = 45.

Example 11: x / 3 = 7.5

Problem: Solve x / 3 = 7.5.

Step 1: The variable is divided by 3.

Step 2: Multiply both sides by 3: x = 7.5 * 3.

Step 3: Calculate: x = 22.5.

Check: 22.5 / 3 = 7.5.

Answer: x = 22.5.

How to solve any one-step equation

Step 1: Locate the variable.

Step 2: Identify the operation attached to the variable.

Step 3: Use the inverse operation.

Step 4: Do the same operation to both sides.

Step 5: Check the answer in the original equation.

Two-step equation preview

Grade 6 often focuses on one-step equations first, but some classes introduce simple two-step equations.

A two-step equation has two operations to undo.

Example: 2x + 3 = 17.

Undo addition first: 2x = 14.

Then undo multiplication: x = 7.

Check: 2 * 7 + 3 = 17.

Word problem example 1

Problem: A student had some pencils. She bought 12 more. Now she has 35 pencils. How many did she have before?

Step 1: Let x be the number of pencils she had before.

Step 2: Write the equation: x + 12 = 35.

Step 3: Subtract 12 from both sides: x = 35 - 12.

Step 4: Calculate: x = 23.

Answer: She had 23 pencils before.

Word problem example 2

Problem: A pack has 8 markers. There are 72 markers total. How many packs are there?

Step 1: Let x be the number of packs.

Step 2: Each pack has 8 markers, so write 8x = 72.

Step 3: Divide by 8: x = 72 / 8.

Step 4: Calculate: x = 9.

Answer: There are 9 packs.

Word problem example 3

Problem: A ribbon was cut into 4 equal pieces. Each piece is 6 inches long. How long was the ribbon before it was cut?

Step 1: Let x be the original ribbon length.

Step 2: Cutting into 4 equal pieces means x / 4 = 6.

Step 3: Multiply both sides by 4: x = 6 * 4.

Step 4: Calculate: x = 24.

Answer: The ribbon was 24 inches long.

Tips for accurate solving

Write one step per line. It keeps the work readable.

Circle or underline the operation attached to the variable.

Use inverse operations instead of guessing.

Keep both sides balanced.

Check by substituting the answer into the first equation.

If the answer does not check, go back to the step where the inverse operation was chosen.

Common mistakes

Mistake: Solving x + 9 = 20 by adding 9 again. Fix: subtract 9 because subtraction undoes addition.

Mistake: Solving 5x = 40 by subtracting 5. Fix: divide by 5 because division undoes multiplication.

Mistake: Changing only one side of the equation. Fix: whatever you do to one side, do to the other side.

Mistake: Checking in the last line only. Fix: check in the original equation.

Mixed examples

Example A: x + 15 = 42. Subtract 15: x = 27.

Example B: x - 11 = 30. Add 11: x = 41.

Example C: 6x = 54. Divide by 6: x = 9.

Example D: x / 8 = 5. Multiply by 8: x = 40.

Example E: 3x + 2 = 20. Subtract 2 to get 3x = 18, then divide by 3 to get x = 6.

Practice questions

1. Solve x + 7 = 18.

2. Solve x - 13 = 25.

3. Solve 5x = 45.

4. Solve x / 6 = 8.

5. Solve x + 2.5 = 9.

6. A box holds 9 books. There are 63 books total. Write and solve an equation for the number of boxes.

Practice answers

1. x = 18 - 7 = 11.

2. x = 25 + 13 = 38.

3. x = 45 / 5 = 9.

4. x = 8 * 6 = 48.

5. x = 9 - 2.5 = 6.5.

6. Let x be the number of boxes. 9x = 63, so x = 7 boxes.