Sometimes the variable shows up on both sides, as in $5x + 3 = 2x + 18$. You already know how to solve multi-step equations; the new move is to gather the variable terms together first.
Why it works
Picture a balance with 5 bags and 3 blocks on the left, 2 bags and 18 blocks on the right. Every bag weighs the same unknown amount. Take 2 bags off each side and the scale stays level: 3 bags and 3 blocks balance 18 blocks. That is $3x + 3 = 18$, a two-step equation.
Key idea
Add or subtract the same variable term on both sides so the variable is on only one side. Moving the smaller variable term usually keeps the numbers positive. Then finish as usual.
Worked examples
Worked example: Gather the x terms
Solve $5x + 3 = 2x + 18$.
- Subtract $2x$ from both sides: $3x + 3 = 18$.
- Subtract 3: $3x = 15$.
- Divide by 3: $x = 5$. Check: $25 + 3 = 28$ and $10 + 18 = 28$.
Answer: $x = 5$
Worked example: Negative coefficients
Solve $4 - 3x = 2x - 11$.
- Add $3x$ to both sides: $4 = 5x - 11$.
- Add 11: $15 = 5x$.
- Divide by 5: $x = 3$. Check: $4 - 9 = -5$ and $6 - 11 = -5$.
Answer: $x = 3$
Worked example: Parentheses on both sides
Solve $2(3x - 1) = 4(x + 2)$.
- Distribute: $6x - 2 = 4x + 8$.
- Subtract $4x$: $2x - 2 = 8$.
- Add 2, then divide by 2: $x = 5$. Check: $2(14) = 28$ and $4(7) = 28$.
Answer: $x = 5$
Worked example: Clear the fractions
Solve $\dfrac{x}{2} + 3 = \dfrac{x}{3} + 5$.
- Multiply every term by 6, a common denominator: $3x + 18 = 2x + 30$.
- Subtract $2x$: $x + 18 = 30$.
- Subtract 18: $x = 12$. Check: $6 + 3 = 9$ and $4 + 5 = 9$.
Answer: $x = 12$
Common mistake
Subtracting a variable term on one side and adding it on the other. From $5x + 3 = 2x + 18$, writing $7x + 3 = 18$ adds $2x$ on the left but takes it away on the right, which tips the balance. Whatever you do, do it to both sides: subtracting $2x$ from both gives $3x + 3 = 18$.