Solving Multi-Step Linear Equations: Step-by-Step Guide
Some equations need more than two steps. They might have parentheses, fractions, or x on both sides. The plan is: get rid of parentheses and fractions, simplify each side, move all the x terms to one side and the plain numbers to the other, and then divide to find x.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Remove parentheses and fractions. Use the distributive property to multiply out any parentheses. If there are fractions, multiply every term on both sides by a number that clears all the denominators.\[k(ax+b) = kax + kb\]
- Combine like terms on each side. Clean up the left side and the right side separately. Add the x terms together, and add the plain numbers together.\[ax + bx + c = (a+b)x + c\]
- Move the x terms to one side. Add or subtract an x term on both sides so that all the x terms are on one side and all the plain numbers are on the other.\[ax + b = cx + d \;\Rightarrow\; (a-c)x = d - b\]
- Get x by itself. Divide both sides by the number in front of x.\[x = \frac{d-b}{a-c}\]
- Check your answer. Put your answer into the original equation (the one you started with). Both sides should be equal.\[\text{LHS}(x) \stackrel{?}{=} \text{RHS}(x)\]
Worked example
Solve: 2(x + 3) - 4 = 5x - 7
Step 1: Distribute the 2: 2(x + 3) = 2x + 6. The equation is now 2x + 6 - 4 = 5x - 7.
Step 2: Combine like terms on the left: 6 - 4 = 2, so we have 2x + 2 = 5x - 7.
Step 3: Subtract 2x from both sides: 2 = 3x - 7. Then add 7 to both sides: 9 = 3x.
Step 4: Divide both sides by 3: x = 3.
Step 5: The answer is x = 3.
Check: Put x = 3 into the original. Left side: 2(3 + 3) - 4 = 12 - 4 = 8. Right side: 5(3) - 7 = 15 - 7 = 8. They match.
Tips and common mistakes
- Work in order: distribute, then combine like terms, then move the x terms. Skipping around leads to mistakes.
- When you distribute a negative number, every sign inside the parentheses flips.
- To clear fractions, multiply every term on both sides, not just one.
- Move the x terms to the side where the number in front of x will stay positive. It makes fewer sign mistakes.
- If the x terms disappear and you get something true like 5 = 5, every number is a solution.
- If the x terms disappear and you get something false like 5 = 7, there is no solution.
- Check your answer in the original equation, not in one of the later steps.