Equations with x on both sides
What you'll learn
Solve any linear equation by keeping the balance: distribute, collect the x terms, and undo what's left.
In pre-algebra you solved x + 5 = 12 by undoing one step. Real equations are messier: 3x + 2 = x + 8 has x on both sides. The tool that handles every case is the same one that handled the first: an equation is a balance.
The balance
Both pans weigh the same — that's what the equals sign claims. So any move is legal as long as you make it on both pans: remove an x-tile from each side, remove two unit tiles from each side, or split everything in half. The scale never tips.
The four-step routine
Every linear equation gives in to the same sequence:
| Step | Move | 3(x + 1) = x + 7 becomes |
|---|---|---|
| 1 | Distribute — clear the parentheses | 3x + 3 = x + 7 |
| 2 | Collect x — subtract the smaller x-side from both | 2x + 3 = 7 |
| 3 | Isolate the x term — subtract 3 from both | 2x = 4 |
| 4 | Divide by the coefficient | x = 2 |
Check it: 3(2 + 1) = 9 and 2 + 7 = 9. Both pans agree — the solution is right. Checking costs ten seconds and catches almost every slip.
When strange things happen
Follow the routine on 2(x + 3) = 2x + 6 and the x's vanish: 6 = 6. A statement that's always true means every x works (the two sides were the same expression in disguise). If instead you reach something false like 6 = 9, no x works. These aren't errors — they're answers.
Why this matters
Everything ahead — systems, quadratics, and every formula you'll ever rearrange — reduces at its final step to this routine. Master the balance and the rest of algebra is about getting equations into this shape.
Check your understanding
Question 1 of 2
Solve: 5x − 4 = 2x + 11