Pre-algebra and algebraGrades 8–9
Multi-step equations worksheet with answer key
Four equation shapes appear on this sheet: a(x + b), like terms on one side, variables on both sides, and distribution with variables on both sides. Negatives are on by default. Answer key included.
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On this page
When one inverse is not enough
Multi-step equations ask a student to tidy the equation before undoing anything. There may be parentheses to expand, like terms to merge, or x on both sides of the equals sign. Eighth and ninth graders spend much of the year here, and a sheet of varied examples builds the habit of looking first and acting second.
This generator draws from four patterns:
- Distribute: a(x + b) = c
- Combine like terms: an x-term, a constant, and another x-term on the left, equal to a number
- Variables on both sides: ax + b = cx + d
- Distribute and variables on both sides: a(x + b) = cx + d
The sheet preset has negative numbers on, so some constants are subtracted and some solutions are negative. The header lists the strategy in order: distribute, combine like terms, collect the variable on one side, solve, and check.
Settings in practice
- Equation type. The page is set to Multi-step. Choose Mixed to add one-step and two-step equations for cumulative review.
- Include negative numbers. Leave on for a realistic challenge. Turn off for a first pass, when the focus is on the sequence of steps and not on signs.
- Use different letters. Helpful when students tend to assume every variable is x.
- Number of problems. From 6 to 30. Equations print two to a row with space beside each for working, so about 16 fit comfortably on a page.
Because the generator picks one of the four shapes at random, a sheet is a mixture, so expect each pattern to appear several times in a set of 16.
Worked example: 4(x + 3) = 2x + 22
- Distribute. Multiply each term in the parentheses by 4: 4x + 12 = 2x + 22.
- Collect the variable. Subtract 2x from both sides: 2x + 12 = 22.
- Isolate the term. Subtract 12 from both sides: 2x = 10.
- Solve. Divide by 2: x = 5.
- Check. Left side: 4 × (5 + 3) = 4 × 8 = 32. Right side: 2 × 5 + 22 = 10 + 22 = 32. They match.
A second pattern, variables on both sides: 5x + 3 = 2x + 21. Subtract 2x from both sides to get 3x + 3 = 21, subtract 3 to get 3x = 18, and divide by 3 to find x = 6. Check: 5 × 6 + 3 = 33 and 2 × 6 + 21 = 33.
Which variable to move is a choice. Moving the smaller coefficient, as above, avoids negative coefficients, but moving the other term gives the same answer.
Mistakes the key will expose
- Distributing to only the first term. Writing 4x + 3 for 4(x + 3) is the most common error. The key answer will not match, and checking by substitution shows why.
- Sign errors when moving terms. Subtracting 2x from one side only, or adding it where subtraction was needed, leaves a wrong coefficient.
- Combining unlike terms. Merging 3x and 5 into 8x. Only like terms combine.
- Stopping at 2x = 10. The equation is not finished until the variable stands alone.
When many answers are off by a sign, return to the integer rules with the integers worksheet.
Building up to this page
Students who are not ready should step back to the two-step equations worksheet. Distribution itself has its own practice in the distributive property worksheets, which is a good warm-up before attempting this sheet.