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Solve equations requiring two steps
Learn step-by-step with practice exercises built right in.
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!
A two-step equation requires TWO inverse operations to solve.
Examples:
Goal: Isolate the variable (get x by itself)
Inverse operations undo each other:
Addition ↔ Subtraction Multiplication ↔ Division
To isolate a variable, use inverse operations to undo what's been done to it!
Order matters! Work backwards from PEMDAS:
Step 1: Undo addition or subtraction (work backwards from order of operations) Undo multiplication or division
Solve for x: 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
Solve for n: 3n - 7 = 8
Avoid these 3 frequent errors
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Think: Reverse order of operations!
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
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
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
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
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!
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!
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
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
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
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!
Shopping: A shirt costs 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 20. After how many weeks will you have $140?
Let w = weeks 15w + 20 = 140 15w = 120 w = 8
8 weeks
Phrase: "5 more than 3 times a number is 20"
Translate:
Equation: 3x + 5 = 20
Solve: 3x = 15 x = 5
Addition:
Subtraction:
Multiplication:
Division:
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!
❌ Mistake 1: Wrong order of operations
❌ Mistake 2: Only operating on one side
❌ Mistake 3: Sign errors with negatives
❌ Mistake 4: Not checking answer
❌ Mistake 5: Forgetting to flip with fractions
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)
One-Step: x + 5 = 12
Two-Step: 2x + 5 = 13
Two-step builds on one-step skills!
Reading the problem:
Solving the equation:
Standard Form: ax + b = c
Solution Steps:
Example: 3x + 7 = 22
Remember:
Tip 1: Write out every step
Tip 2: Keep equals signs aligned
Tip 3: Check with substitution
Tip 4: Practice translating words
Tip 5: Draw diagrams when helpful
Two-step equations require two inverse operations to solve:
Standard form: ax + b = c
Solution method:
Key principles:
Applications:
Skills needed:
Mastering two-step equations prepares you for multi-step equations and algebraic problem solving!
Step 1: Add 7 to both sides (undo subtraction). 3n - 7 + 7 = 8 + 7 3n = 15
Step 2: Divide both sides by 3 (undo multiplication). 3n/3 = 15/3 n = 5
Step 3: Check. 3(5) - 7 = 15 - 7 = 8 ✓
Answer: n = 5
Solve for y: y/4 + 3 = 7
Step 1: Subtract 3 from both sides. y/4 + 3 - 3 = 7 - 3 y/4 = 4
Step 2: Multiply both sides by 4 (undo division). 4 × (y/4) = 4 × 4 y = 16
Step 3: Check. 16/4 + 3 = 4 + 3 = 7 ✓
Answer: y = 16
Solve for x: -5x + 12 = -3
Step 1: Subtract 12 from both sides. -5x + 12 - 12 = -3 - 12 -5x = -15
Step 2: Divide both sides by -5. -5x/-5 = -15/-5 x = 3
Note: Negative divided by negative = positive
Step 3: Check. -5(3) + 12 = -15 + 12 = -3 ✓
Answer: x = 3
A cell phone plan costs 0.10 per text message. If your bill was $37, how many text messages did you send?
Step 1: Write an equation. Let t = number of texts Cost = 25 + 0.10t Equation: 25 + 0.10t = 37
Step 2: Subtract 25 from both sides. 0.10t = 12
Step 3: Divide both sides by 0.10. t = 12 ÷ 0.10 t = 120
Step 4: Check. 25 + 0.10(120) = 25 + 12 = 37 ✓
Answer: You sent 120 text messages