How to Solve Multi-Step Equations? (+FREE Worksheet!)
How to Solve Multi-Step Equations
A multi-step equation just means you’ll do a few moves, distribute, combine like terms, then undo addition and multiplication, before \(x\) stands alone. Learn the order and these stop feeling scary. We’ll work plenty together, with a solver, drills, and a worksheet maker one tap away.
Solve Multi-Step Equations: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Simplify each sideDistribute and combine like terms before moving variables.
- Collect variablesUse inverse operations to get variable terms on one side and constants on the other.
- Check in the originalSubstitute the solution into the original equation, the simplified line.
Worked examples
Two-step equation
- Subtract 5 from both sides.
- Divide both sides by 3.
- Check 3(5) + 5 = 20.
Variables on both sides
- Subtract 2x from both sides.
- Add 7 to both sides.
- Divide by 2.
Try one before moving on
Solve Multi-Step Equations: pop-up practice
If a one-step equation is a single move, a multi-step equation is a short combo: maybe a parenthesis to distribute, some like terms to combine, variables on both sides to gather, and then the usual “undo” steps to free \(x\). It looks busier, but every move is one you already know. The secret is doing them in a sensible order, and we’ll practice that order until it feels automatic. Every move here is one you already know from one-step equations; we’re just chaining a few together.
What Is a Multi-Step Equation?
A multi-step equation is an equation that takes two or more operations to isolate the variable, for example \(3(x-2)+4=16\). You solve it by simplifying each side first (distribute, then combine like terms), then gathering the variable on one side and undoing operations, all while keeping the equation balanced.
The reliable order of moves:
- Distribute to clear any parentheses.
- Combine like terms on each side.
- Move the variables to one side (add/subtract).
- Undo addition/subtraction, then multiplication/division.
From Tangle to Solution, Step by Step
Here’s the order in action on \(3(x – 2) + 4 = 16\):
Distribute
\(\Rightarrow 3x-6+4=16\)
Combine
\(\Rightarrow 3x-2=16\)
Unwrap
\(\Rightarrow 3x=18\Rightarrow x=\) 6
Worked Examples
Tidy up, then unwrap, each card follows the equation down to its solution.
Example A, Variables on both sides
Solve \(2x + 5 = 15 – x\).
- Add \(x\) to both sides: \(3x + 5 = 15\).
- Subtract 5: \(3x = 10\).
- Divide by 3: \(x = \tfrac{10}{3}\), a fraction is a fine answer.
Answer: \(x = \tfrac{10}{3}\)
Example B, Distribute first
Solve \(3(x – 2) + 4 = 16\).
- Distribute: \(3x – 6 + 4 = 16\).
- Combine: \(3x – 2 = 16\); add 2: \(3x = 18\).
- Divide by 3: \(x = 6\).
Answer: \(x = 6\)
Example C, Gather the variables
Solve \(7x – 4 = 3x + 16\).
- Subtract \(3x\): \(4x – 4 = 16\).
- Add 4: \(4x = 20\); divide: \(x = 5\).
- Check: \(7(5)-4 = 31 = 3(5)+16\) ✓
Answer: \(x = 5\)
Example G, Clear the fraction first
Solve \(\dfrac{x}{2} + 3 = 7\).
- Multiply every term by 2: \(x + 6 = 14\).
- Subtract 6: \(x = 8\).
- Clearing the denominator early keeps the rest clean.
Answer: \(x = 8\)
Example D, No solution
Solve \(2x + 1 = 2x + 5\).
- Subtract \(2x\): \(1 = 5\).
- The variable vanished and left a false statement.
- So there’s no solution.
Answer: no solution
Example E, Infinitely many
Solve \(3(x + 1) = 3x + 3\).
- Distribute: \(3x + 3 = 3x + 3\).
- Both sides are identical, true for every \(x\).
- So there are infinitely many solutions.
Answer: infinitely many
Example F, A real-world setup
A taxi charges $3 plus $2 per mile. For a $17 ride: solve \(2m + 3 = 17\).
- Subtract 3: \(2m = 14\).
- Divide by 2: \(m = 7\).
- Name the unknown and the word problem becomes a multi-step equation.
Answer: 7 miles
Slip-Ups That Cost Easy Points
- Distributing to only the first term. \(3(x-2)\) is \(3x – 6\), not \(3x – 2\). Multiply the outside number by every term inside.
- Forgetting to do it to both sides. Subtract 5 from the left? Subtract 5 from the right too. That’s what keeps the equation true.
- Sign errors when moving terms. Moving \(+3x\) across makes it \(-3x\). Write the step out instead of doing it in your head.
- Panicking at “no solution” or “all reals.” If the variable disappears, a false line (\(1=5\)) means no solution and a true line (\(3=3\)) means infinitely many. That’s information, not a mistake.
Your Turn: Solve These
Work each by hand, then reveal the answers. Stuck on one? The step-by-step solver shows every move.
- \(5x – 7 = 3x + 9\)
- \(2(x – 4) = 10\)
- \(6x + 4 = 2x – 12\)
- \(-3(x – 5) = 9\)
- \(\dfrac{x}{3} + 2 = 5\)
- \(10 – 2x = 4x – 8\)
- \(4x + 2 = 4x – 7\)
Show answers
- \(\color{blue}{x=8}\)
- \(\color{blue}{x=9}\)
- \(\color{blue}{x=-4}\)
- \(\color{blue}{x=2}\)
- \(\color{blue}{x=9}\)
- \(\color{blue}{x=3}\)
- \(\color{blue}{\text{no solution } (2=-7\text{ is false})}\)
Make Your Own Equations Worksheet
Generate fresh multi-step equations with a full answer key, print or save as a PDF.
Frequently Asked Questions
What is a multi-step equation?
A multi-step equation needs two or more operations to get the variable alone. You may have to distribute, combine like terms or move variables from both sides before the final division. For example, 2(x + 4) = 18 takes two steps: distribute to get 2x + 8 = 18, subtract 8 to get 2x = 10, then divide by 2, so x = 5.
How do you solve multi-step equations?
Simplify each side first by distributing and combining like terms, then use inverse operations to isolate the variable. Undo addition or subtraction before multiplication or division. For 3x + 2x − 4 = 21, combine to 5x − 4 = 21, add 4 to get 5x = 25, and divide by 5, so x = 5. Check by substituting 5 into the original equation.
How do you solve multi-step equations with variables on both sides?
Collect the variable terms on one side and the constants on the other. For 7x + 3 = 4x + 15, subtract 4x from both sides to get 3x + 3 = 15. Subtract 3 to get 3x = 12, then divide by 3, so x = 4. Check: 7(4) + 3 = 31 and 4(4) + 15 = 31, so both sides match.
How do you solve multi-step equations with fractions?
Multiply every term by the least common denominator to clear the fractions, then solve normally. For x/2 + x/3 = 10, multiply by 6 to get 3x + 2x = 60. That gives 5x = 60, so x = 12. Check: 12/2 + 12/3 = 6 + 4 = 10. Clearing fractions first keeps the arithmetic simple and reduces mistakes.
What grade level is multi-step equations?
Multi-step equations are usually taught in 7th and 8th grade and then reviewed in Algebra 1. Seventh graders solve equations like 3x + 5 = 20, and eighth graders work with distributing and variables on both sides, such as 5(x − 1) = 2x + 7. A solid grip on integer operations and the distributive property makes this topic much easier.
Are multi-step equations hard?
They feel hard at first because there are several moves to track, but they become routine once you follow the same order every time: simplify each side, collect variables, isolate, then check. Most errors come from sign slips, such as distributing −2(x − 3) as −2x − 6 instead of −2x + 6. Writing each step on its own line helps a lot.
Related Topics
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