Solving Multi-Step Equations

In Grade 6, you solved one-step equations. Now you're ready for equations that require multiple steps to solve! These equations combine the operations you know, requiring you to use inverse operations strategically.

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What Is a Multi-Step Equation?

A multi-step equation requires two or more steps to isolate the variable.

Examples:

• 2x + 5 = 13 (two steps: subtract, then divide)
• 3(x - 4) = 15 (three steps: distribute, add, divide)
• 5x - 3 = 2x + 9 (variables on both sides!)

Goal: Get the variable alone on one side of the equation.

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The Strategy: Work Backwards!

Think of equations like a wrapped present. To get to the variable inside, you need to unwrap the layers in reverse order!

Order of Operations (PEMDAS): Parentheses, Exponents, Multiply/Divide, Add/Subtract

Solving Strategy (Reverse PEMDAS):

1. Simplify (combine like terms, distribute)
2. Undo addition/subtraction (add/subtract on both sides)
3. Undo multiplication/division (multiply/divide on both sides)

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Two-Step Equations

Example 1: Addition Then Division

Solve: 2x + 5 = 13

Step 1: Subtract 5 from both sides (undo addition) 2x + 5 - 5 = 13 - 5 2x = 8

Step 2: Divide both sides by 2 (undo multiplication) 2x ÷ 2 = 8 ÷ 2 x = 4

Step 3: Check your answer 2(4) + 5 = 8 + 5 = 13 ✓

Answer: x = 4

Example 2: Subtraction Then Division

Solve: 3x - 7 = 14

Step 1: Add 7 to both sides 3x - 7 + 7 = 14 + 7 3x = 21

Step 2: Divide both sides by 3 3x ÷ 3 = 21 ÷ 3 x = 7

Check: 3(7) - 7 = 21 - 7 = 14 ✓

Answer: x = 7

Example 3: Division Then Subtraction

Solve: x/4 + 3 = 8

Step 1: Subtract 3 from both sides x/4 + 3 - 3 = 8 - 3 x/4 = 5

Step 2: Multiply both sides by 4 x/4 × 4 = 5 × 4 x = 20

Check: 20/4 + 3 = 5 + 3 = 8 ✓

Answer: x = 20

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Equations with the Distributive Property

When parentheses appear, distribute first!

Example 1: Distribute Then Solve

Solve: 2(x + 3) = 16

Step 1: Distribute the 2 2x + 6 = 16

Step 2: Subtract 6 from both sides 2x = 10

Step 3: Divide both sides by 2 x = 5

Check: 2(5 + 3) = 2(8) = 16 ✓

Answer: x = 5

Example 2: Negative Distribution

Solve: -3(x - 4) = 21

Step 1: Distribute -3 -3x + 12 = 21

Step 2: Subtract 12 from both sides -3x = 9

Step 3: Divide both sides by -3 x = -3

Check: -3(-3 - 4) = -3(-7) = 21 ✓

Answer: x = -3

Example 3: Distribution Plus More

Solve: 4(x + 2) - 5 = 19

Step 1: Distribute the 4 4x + 8 - 5 = 19

Step 2: Combine like terms on left 4x + 3 = 19

Step 3: Subtract 3 from both sides 4x = 16

Step 4: Divide both sides by 4 x = 4

Answer: x = 4

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Combining Like Terms First

Always simplify before solving!

Example 1: Combine on One Side

Solve: 3x + 2x + 7 = 22

Step 1: Combine like terms (3x + 2x) 5x + 7 = 22

Step 2: Subtract 7 5x = 15

Step 3: Divide by 5 x = 3

Answer: x = 3

Example 2: Combine on Both Sides

Solve: 2x + 5 + 3x = 4 + 11

Step 1: Combine left side (2x + 3x) 5x + 5 = 4 + 11

Step 2: Combine right side (4 + 11) 5x + 5 = 15

Step 3: Subtract 5 5x = 10

Step 4: Divide by 5 x = 2

Answer: x = 2

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Variables on Both Sides

When the variable appears on both sides, get all variables on one side first!

Example 1: Basic Variables on Both Sides

Solve: 5x = 3x + 10

Step 1: Subtract 3x from both sides 5x - 3x = 3x + 10 - 3x 2x = 10

Step 2: Divide by 2 x = 5

Check: 5(5) = 3(5) + 10 → 25 = 15 + 10 → 25 = 25 ✓

Answer: x = 5

Example 2: With Constants Too

Solve: 4x + 3 = 2x + 11

Step 1: Subtract 2x from both sides 4x - 2x + 3 = 2x - 2x + 11 2x + 3 = 11

Step 2: Subtract 3 from both sides 2x = 8

Step 3: Divide by 2 x = 4

Answer: x = 4

Example 3: Move Variables to Left

Solve: 7 + 3x = 5x - 9

Step 1: Subtract 3x from both sides 7 = 2x - 9

Step 2: Add 9 to both sides 16 = 2x

Step 3: Divide by 2 8 = x or x = 8

Answer: x = 8

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Equations with Fractions

Clear fractions by multiplying by the LCD!

Example 1: One Fraction

Solve: (x + 3)/2 = 7

Step 1: Multiply both sides by 2 2 × (x + 3)/2 = 7 × 2 x + 3 = 14

Step 2: Subtract 3 x = 11

Answer: x = 11

Example 2: Multiple Fractions

Solve: x/3 + x/6 = 5

Step 1: Find LCD (6) and multiply everything by it 6 × (x/3) + 6 × (x/6) = 6 × 5 2x + x = 30

Step 2: Combine like terms 3x = 30

Step 3: Divide by 3 x = 10

Answer: x = 10

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Equations with Decimals

You can clear decimals by multiplying by powers of 10!

Example: Clear Decimals

Solve: 0.5x + 1.2 = 3.7

Method 1: Work with decimals 0.5x = 2.5 x = 5

Method 2: Clear decimals (multiply by 10) 10(0.5x) + 10(1.2) = 10(3.7) 5x + 12 = 37 5x = 25 x = 5

Answer: x = 5

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Step-by-Step Strategy

Step 1: Simplify Each Side

• Distribute if needed
• Combine like terms

Step 2: Get Variables on One Side

• Add or subtract to move variables to one side

Step 3: Get Constants on the Other Side

• Add or subtract to move numbers to the other side

Step 4: Solve for the Variable

• Multiply or divide to get the variable alone

Step 5: Check Your Answer

• Substitute back into original equation

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Real-World Applications

Shopping

Problem: You have $50. You buy a shirt for$18 and some socks for $4 each. How many pairs of socks can you buy?

Equation: 18 + 4x = 50

Solution: 4x = 32 x = 8

Answer: 8 pairs of socks

Geometry

Problem: The perimeter of a rectangle is 40 cm. The length is 3 cm more than twice the width. Find the width.

Let w = width, then length = 2w + 3

Equation: 2w + 2(2w + 3) = 40

Solution: 2w + 4w + 6 = 40 6w + 6 = 40 6w = 34 w = 34/6 = 17/3 or about 5.67 cm

Answer: Width = 17/3 cm

Temperature

Problem: The formula F = 9C/5 + 32 converts Celsius to Fahrenheit. If it's 77°F, what is the Celsius temperature?

Equation: 77 = 9C/5 + 32

Solution: 45 = 9C/5 225 = 9C C = 25

Answer: 25°C

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Common Mistakes to Avoid

❌ Mistake 1: Not distributing to all terms

• Wrong: 2(x + 3) = 2x + 3
• Right: 2(x + 3) = 2x + 6

❌ Mistake 2: Only doing operation on one side

• Wrong: 2x + 5 = 13 → 2x = 13 (forgot to subtract 5 from right side!)
• Right: 2x + 5 = 13 → 2x = 8

❌ Mistake 3: Sign errors when moving variables

• Wrong: 5x = 3x + 10 → 2x = 10 + 3x
• Right: 5x = 3x + 10 → 2x = 10

❌ Mistake 4: Not combining like terms first

• Simplify before you start solving!

❌ Mistake 5: Forgetting to check

• Always substitute your answer back into the original equation!

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Practice Tips

Tip 1: Write neatly and show all steps

• Sloppy work leads to errors
• Each line should show one operation

Tip 2: Keep equation balanced

• Whatever you do to one side, do to the other
• Think of a balanced scale

Tip 3: Work in order

• Simplify → Variables to one side → Constants to other side → Solve

Tip 4: Use inverse operations

• Addition ↔ Subtraction
• Multiplication ↔ Division

Tip 5: Check your answer every time

• Plug it back in
• Both sides should be equal

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Special Cases

Infinite Solutions

Some equations are true for ALL values!

Example: 2(x + 3) = 2x + 6

When you simplify: 2x + 6 = 2x + 6 (always true!)

Answer: All real numbers (infinite solutions)

No Solution

Some equations are NEVER true!

Example: x + 5 = x + 3

When you simplify: 5 = 3 (false!)

Answer: No solution

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Summary

Multi-step equations require multiple operations to solve.

Strategy:

1. Simplify (distribute, combine like terms)
2. Move variables to one side
3. Move constants to other side
4. Divide or multiply to isolate variable
5. Check your answer!

Key Skills:

• Distributive property
• Combining like terms
• Inverse operations
• Working with variables on both sides

Remember: Whatever you do to one side, you MUST do to the other side. Keep the equation balanced!

Master multi-step equations and you're ready for inequalities, systems of equations, and advanced algebra!