Linear Equations • Topic 1 of 4

Solving in One Variable

A one-variable linear equation is solved by undoing operations in reverse order. Simplify each side first — distribute brackets and combine like terms — then move all variable terms to one side and all constants to the other, and finally divide by the coefficient of the variable. Whatever you do to one side you must do to the other; that is what keeps the equation balanced. Always check by substituting your answer back into the original; the GRE builds fast, and a 15-second check catches a dropped sign. Two special cases matter: if the variables cancel and leave a true statement like 5 = 5, every number works (infinitely many solutions); if they leave a false statement like 5 = 8, no number works (no solution).

✅ Solved examples

1. Solve 3x + 7 = 22.
Subtract 7: 3x = 15. Divide by 3: x = 5.
2. Solve 5x − 4 = 2x + 11.
Gather variables left, constants right: 5x − 2x = 11 + 4, so 3x = 15, x = 5.
3. Solve 2(x − 3) = 4x + 2.
Distribute: 2x − 6 = 4x + 2. Subtract 2x: −6 = 2x + 2. Subtract 2: −8 = 2x, so x = −4.
4. Solve 4(x + 1) = 4x + 4 and state the number of solutions.
Expand: 4x + 4 = 4x + 4. Subtract 4x: 4 = 4, always true — infinitely many solutions (every real x).

✏️ Practice — try these, take hints as needed

1. Solve 6x − 5 = 19.
Add 5 to both sides.
6x = 24.
Divide by 6.
x = 4
2. Solve 7x + 2 = 3x + 18.
Move variables one side.
7x − 3x = 18 − 2.
4x = 16.
x = 4
3. Solve 3(2x − 1) = 2x + 13.
Distribute first.
6x − 3 = 2x + 13.
4x = 16.
x = 4
4. Solve 5x + 8 = 5x − 2 and state the number of solutions.
Subtract 5x from both sides.
8 = −2 is false.
No value can work.
No solution
5. Solve −2(x − 4) = 10.
Distribute the −2.
−2x + 8 = 10.
−2x = 2.
x = −1

📝 Topic test — 8 questions

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