Isolating the variable

To "isolate" a variable means to get it alone on one side of the equals sign. The fastest path is almost always:

1. Combine like terms on each side first.
2. Move variables to one side using addition / subtraction.
3. Move constants to the other side.
4. Divide by the coefficient.

Worked example

Solve 5x − 3 = 2x + 12.

• Step 2: subtract 2x from both sides → 3x − 3 = 12
• Step 3: add 3 → 3x = 15
• Step 4: divide by 3 → x = 5

Plug back in: 5(5) − 3 = 22 and 2(5) + 12 = 22. ✅

Common trap

If the variable disappears (e.g. you end up with 7 = 7), every number is a solution. If you end up with a contradiction (e.g. 3 = 5), there is no solution.