Solving Two-Step Equations

How do you solve equations that require two operations? Two-step equations build on one-step equations and are essential for solving real-world problems!

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What Is a Two-Step Equation?

A two-step equation requires TWO inverse operations to solve.

Examples:

• 2x + 5 = 13 (multiply and add)
• 3x - 7 = 11 (multiply and subtract)
• x/4 + 3 = 8 (divide and add)

Goal: Isolate the variable (get x by itself)

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Review: Inverse Operations

Inverse operations undo each other:

Addition ↔ Subtraction Multiplication ↔ Division

To isolate a variable, use inverse operations to undo what's been done to it!

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General Strategy

Order matters! Work backwards from PEMDAS:

Step 1: Undo addition or subtraction (work backwards from order of operations) Step 2: Undo multiplication or division

Think: Reverse order of operations!

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Solving: Addition First

Example: 2x + 5 = 13

Step 1: Subtract 5 from both sides 2x + 5 - 5 = 13 - 5 2x = 8

Step 2: Divide both sides by 2 2x/2 = 8/2 x = 4

Check: 2(4) + 5 = 8 + 5 = 13 ✓

Answer: x = 4

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Solving: Subtraction First

Example: 3x - 7 = 11

Step 1: Add 7 to both sides 3x - 7 + 7 = 11 + 7 3x = 18

Step 2: Divide both sides by 3 3x/3 = 18/3 x = 6

Check: 3(6) - 7 = 18 - 7 = 11 ✓

Answer: x = 6

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Solving: Division Then Addition

Example: 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 4 × (x/4) = 5 × 4 x = 20

Check: 20/4 + 3 = 5 + 3 = 8 ✓

Answer: x = 20

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Solving: Division Then Subtraction

Example: x/5 - 2 = 4

Step 1: Add 2 to both sides x/5 - 2 + 2 = 4 + 2 x/5 = 6

Step 2: Multiply both sides by 5 x = 30

Check: 30/5 - 2 = 6 - 2 = 4 ✓

Answer: x = 30

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When Variable Is Negative

Example: -2x + 6 = 14

Step 1: Subtract 6 from both sides -2x + 6 - 6 = 14 - 6 -2x = 8

Step 2: Divide both sides by -2 -2x/-2 = 8/-2 x = -4

Check: -2(-4) + 6 = 8 + 6 = 14 ✓

Answer: x = -4

Remember: Negative ÷ Negative = Positive!

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When Coefficient Is a Fraction

Example: (1/2)x + 4 = 9

Step 1: Subtract 4 from both sides (1/2)x = 5

Step 2: Multiply both sides by 2 (reciprocal of 1/2) x = 10

Check: (1/2)(10) + 4 = 5 + 4 = 9 ✓

Answer: x = 10

Tip: Multiply by the reciprocal to undo fraction multiplication!

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When Constant Is Negative

Example: 5x - 8 = -13

Step 1: Add 8 to both sides 5x - 8 + 8 = -13 + 8 5x = -5

Step 2: Divide both sides by 5 x = -1

Check: 5(-1) - 8 = -5 - 8 = -13 ✓

Answer: x = -1

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Variable on the Right Side

Example: 12 = 4 + 2x

Can solve as is, or flip the equation!

Method 1: Solve as is 12 - 4 = 4 + 2x - 4 8 = 2x 4 = x

Method 2: Flip equation 2x + 4 = 12 (now solve normally)

Answer: x = 4

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More Complex Examples

Example 1: 7x + 15 = 50

Step 1: Subtract 15 7x = 35

Step 2: Divide by 7 x = 5

Answer: x = 5

Example 2: x/3 - 6 = -2

Step 1: Add 6 x/3 = 4

Step 2: Multiply by 3 x = 12

Answer: x = 12

Example 3: -4x + 20 = 4

Step 1: Subtract 20 -4x = -16

Step 2: Divide by -4 x = 4

Answer: x = 4

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Checking Your Answer

ALWAYS check by substituting back!

Example: Solution is x = 3 for equation 2x + 1 = 7

Check: 2(3) + 1 = 7 6 + 1 = 7 7 = 7 ✓

If both sides equal, your answer is correct!

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Real-World Applications

Shopping: A shirt costs $5 more than twice the price of a hat. Together they cost$35. Find hat price.

Let h = hat price 2h + 5 = 35 2h = 30 h = 15

Hat costs $15

Age Problem: Maria is 3 years older than twice Juan's age. Maria is 19. How old is Juan?

Let j = Juan's age 2j + 3 = 19 2j = 16 j = 8

Juan is 8 years old

Savings: You save $15 per week plus an initial$20. After how many weeks will you have $140?

Let w = weeks 15w + 20 = 140 15w = 120 w = 8

8 weeks

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Writing Equations from Words

Phrase: "5 more than 3 times a number is 20"

Translate:

• "a number" → x
• "3 times a number" → 3x
• "5 more than" → + 5
• "is" → =

Equation: 3x + 5 = 20

Solve: 3x = 15 x = 5

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Common Phrases to Equations

Addition:

• "more than" → +
• "increased by" → +
• "sum of" → +

Subtraction:

• "less than" → -
• "decreased by" → -
• "difference" → -

Multiplication:

• "times" → ×
• "product of" → ×
• "twice" → 2×

Division:

• "divided by" → ÷
• "quotient" → ÷

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Step-by-Step Process

For any two-step equation:

Step 1: Identify operations on the variable Step 2: Undo addition/subtraction first Step 3: Undo multiplication/division second Step 4: Check your answer

Key: Whatever you do to one side, do to the other!

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Common Mistakes to Avoid

❌ Mistake 1: Wrong order of operations

• Wrong: Divide first in 2x + 5 = 13
• Right: Subtract first, then divide

❌ Mistake 2: Only operating on one side

• Wrong: 2x + 3 = 9 → 2x = 9
• Right: 2x + 3 = 9 → 2x + 3 - 3 = 9 - 3

❌ Mistake 3: Sign errors with negatives

• Be careful: -2x ÷ -2 = x (not -x)
• Negative ÷ Negative = Positive

❌ Mistake 4: Not checking answer

• Always substitute back!
• Catches arithmetic errors

❌ Mistake 5: Forgetting to flip with fractions

• To undo ×(1/2), multiply by 2 (reciprocal)

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Special Cases

No variable after solving:

Example: 2x + 3 = 2x + 7

Subtract 2x from both sides: 3 = 7 (FALSE!)

This means NO SOLUTION (inconsistent equation)

Always true:

Example: 3x + 2 = 3x + 2

Subtract 3x from both sides: 2 = 2 (TRUE!)

This means INFINITE SOLUTIONS (identity)

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Comparison: One-Step vs Two-Step

One-Step: x + 5 = 12

• One operation to undo
• x = 7

Two-Step: 2x + 5 = 13

• Two operations to undo
• First subtract 5, then divide by 2
• x = 4

Two-step builds on one-step skills!

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Problem-Solving Strategy

Reading the problem:

1. Identify what the variable represents
2. Look for key words (more than, less than, times, etc.)
3. Write the equation
4. Solve using two-step method
5. Check in context

Solving the equation:

1. Undo addition/subtraction
2. Undo multiplication/division
3. Check your answer
4. Write answer with units if applicable

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Quick Reference

Standard Form: ax + b = c

Solution Steps:

1. Subtract (or add) b from both sides
2. Divide (or multiply) both sides by a

Example: 3x + 7 = 22

1. 3x = 15 (subtract 7)
2. x = 5 (divide by 3)

Remember:

• Work backwards from order of operations
• Do the same thing to both sides
• Always check your answer!

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Practice Tips

Tip 1: Write out every step

• Don't skip steps mentally
• Helps avoid errors

Tip 2: Keep equals signs aligned

• Makes it easier to read
• Less likely to make mistakes

Tip 3: Check with substitution

• Plug answer back into original equation
• Both sides should be equal

Tip 4: Practice translating words

• Real problems are in words
• Translating is a key skill

Tip 5: Draw diagrams when helpful

• Visual models aid understanding
• Especially for word problems

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Summary

Two-step equations require two inverse operations to solve:

Standard form: ax + b = c

Solution method:

1. Undo addition/subtraction first
2. Undo multiplication/division second
3. Check your answer

Key principles:

• Do the same operation to both sides
• Work backwards from order of operations
• Addition/subtraction before multiplication/division

Applications:

• Shopping and money problems
• Age problems
• Savings and budget
• Geometry (perimeter, area)
• Distance and rate

Skills needed:

• Inverse operations
• Order of operations
• Working with negatives
• Checking solutions

Mastering two-step equations prepares you for multi-step equations and algebraic problem solving!