Solving linear equations tutorial
A linear equation in one variable contains the unknown $x$ raised only to the first power. Solving it means finding the single value of $x$ that makes both sides equal. The strategy is always the same: use inverse operations to isolate $x$ on one side of the equation.
Core idea
Whatever you do to one side of an equation, you must do to the other. This keeps the equation balanced.
To undo addition → subtract. To undo multiplication → divide. Apply inverse operations until $x$ stands alone.
Two-step equations: $ax + b = c$
Most algebra equations have two steps. The general strategy:
- Undo addition or subtraction first — add or subtract $b$ from both sides.
- Then undo multiplication or division — divide both sides by $a$.
Example: Solve $3x + 5 = 14$
Subtract 5: $3x = 9$
Divide by 3: $x = 3$
Check: $3(3) + 5 = 14$ ✓
Equations with negative coefficients
When the coefficient of $x$ is negative, dividing by a negative number flips a normal arithmetic pattern — but the algebra works the same way.
Example: Solve $-4x + 3 = 11$
Subtract 3: $-4x = 8$
Divide by $-4$: $x = -2$
Check: $-4(-2) + 3 = 8 + 3 = 11$ ✓
Variable on the right side
Sometimes the variable is subtracted: $b - ax = c$. The same inverse-operation strategy applies.
Example: Solve $10 - 2x = 4$
Subtract 10: $-2x = -6$
Divide by $-2$: $x = 3$
Check: $10 - 2(3) = 10 - 6 = 4$ ✓
Fractional coefficients
When $x$ has a fractional coefficient like $\dfrac{x}{3}$, multiply both sides by the denominator to clear the fraction.
Example: Solve $\dfrac{x}{4} + 2 = 5$
Subtract 2: $\dfrac{x}{4} = 3$
Multiply by 4: $x = 12$
Check: $\dfrac{12}{4} + 2 = 3 + 2 = 5$ ✓
General solving checklist
- Simplify each side of the equation separately (distribute, combine like terms).
- Move any constant terms to the side without the variable using addition or subtraction.
- Move any variable terms to one side if needed.
- Divide both sides by the coefficient of $x$ (or multiply to clear a fraction).
- State the value of $x$ and substitute it back to verify.
Try a sample problem
Isolate $x$ and enter the solution. Click Check answer to see the worked solution.