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.
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?