Multi-Step Equations
Level up your algebra skills! Multi-step equations require using inverse operations in the correct order to isolate the variable. Master these equations and you'll be able to solve real-world problems involving money, geometry, and more.
Do this: Read the concept below, then try the quiz or activity.
Concept
Multi-step equations require strategy and order - work backward to isolate the variable!
WHAT ARE MULTI-STEP EQUATIONS? Equations that require more than one operation to solve.
General form: ax + b = c or a(x + b) = c
Example: 3x + 5 = 20 or 2(x - 4) = 10
THE GOLDEN RULE: Whatever you do to one side, do to the other side! This keeps the equation balanced.
STRATEGY: Work BACKWARD (Reverse Order of Operations)
Remember PEMDAS? Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
For SOLVING, reverse it: 1. Undo addition/subtraction FIRST 2. Undo multiplication/division SECOND 3. Undo parentheses (distribute if needed)
TWO-STEP EQUATIONS:
Form: ax + b = c
Example: 3x + 7 = 22
Step 1: Subtract 7 from both sides 3x + 7 - 7 = 22 - 7 3x = 15
Step 2: Divide both sides by 3 3x/3 = 15/3 x = 5
Check: 3(5) + 7 = 15 + 7 = 22 ✓
Form: ax - b = c
Example: 5x - 8 = 17
Step 1: Add 8 to both sides 5x - 8 + 8 = 17 + 8 5x = 25
Step 2: Divide both sides by 5 x = 5
EQUATIONS WITH PARENTHESES:
Method 1: Distribute First
Example: 2(x + 3) = 16
Step 1: Distribute the 2 2x + 6 = 16
Step 2: Subtract 6 2x = 10
Step 3: Divide by 2 x = 5
Method 2: Divide First (when possible)
Example: 4(x - 2) = 20
Step 1: Divide both sides by 4 x - 2 = 5
Step 2: Add 2 x = 7
EQUATIONS WITH FRACTIONS:
Example: (x/4) + 3 = 8
Step 1: Subtract 3 x/4 = 5
Step 2: Multiply by 4 x = 20
EQUATIONS WITH VARIABLES ON BOTH SIDES:
Example: 5x + 3 = 2x + 18
Step 1: Subtract 2x from both sides 3x + 3 = 18
Step 2: Subtract 3 3x = 15
Step 3: Divide by 3 x = 5
EQUATIONS WITH NEGATIVE COEFFICIENTS:
Example: -3x + 7 = -5
Step 1: Subtract 7 -3x = -12
Step 2: Divide by -3 (negative ÷ negative = positive!) x = 4
CHECKING YOUR ANSWER: Always substitute your answer back into the original equation!
Example: If x = 4 in 2x - 5 = 3 Check: 2(4) - 5 = 8 - 5 = 3 ✓
REAL-WORLD APPLICATIONS: - Money: "You have $3 and earn $5 per hour. When will you have $28?" → 5h + 3 = 28 - Geometry: "Perimeter of rectangle is 36, length is 2w + 3" → 2(w) + 2(2w + 3) = 36 - Temperature: "Convert Celsius to Fahrenheit: F = (9/5)C + 32" - Age problems: "John is 5 years older than twice Mary's age. John is 25." → 2m + 5 = 25
Try it
Solve these multi-step equations step by step!
TWO-STEP EQUATIONS (ax + b = c): 1. 4x + 5 = 29 2. 7x - 12 = 30 3. 3x + 8 = 23 4. -2x + 10 = 4 5. 6x - 15 = -3
TWO-STEP EQUATIONS (x/a + b = c): 6. x/3 + 4 = 10 7. x/5 - 2 = 6 8. x/2 + 7 = 15
EQUATIONS WITH PARENTHESES: 9. 2(x + 5) = 24 10. 3(x - 4) = 18 11. 5(2x + 1) = 45 12. -2(x + 7) = -20
EQUATIONS WITH DECIMALS: 13. 1.5x + 3.5 = 11 14. 0.4x - 1.2 = 2.8 15. 2.5x + 7 = 17
MIXED PRACTICE: 16. 9 - 2x = 15 (Hint: Subtract 9 first, then divide by -2) 17. 12 = 3x - 6 18. -4x + 9 = -7 19. (x + 5)/3 = 6 (Multiply both sides by 3 first)
WORD PROBLEMS: 20. Maya has $12 and saves $5 per week. Write and solve an equation to find when she'll have $47.
21. A rectangular garden has a length of 2w + 3 feet and width w feet. If the perimeter is 36 feet, find the width. (Hint: P = 2l + 2w)
22. The formula for converting Celsius to Fahrenheit is F = (9/5)C + 32. If the temperature is 77°F, what is it in Celsius?
23. A phone plan costs $25 per month plus $0.10 per text. If your bill is $34, how many texts did you send?
CHALLENGE: 24. Solve: 4(2x - 3) + 5 = 37
25. Solve: 3x + 7 = x + 19 (variable on both sides)
ERROR ANALYSIS:
26. Find the error in this solution:
2x + 8 = 20
2x = 28 (added 8 instead of subtracting)
x = 14
What is the correct answer?