Solving Linear Equations

Multi-Step Equations

Combine like terms and use inverse operations:

$5x + 3 - 2x = 18$ $3x + 3 = 18$ $3x = 15$ $x = 5$

Variables on Both Sides

Get all variable terms on one side:

$4x + 7 = 2x + 15$ $4x - 2x = 15 - 7$ $2x = 8$ $x = 4$

Distributive Property First

$3(2x - 4) = 18$ $6x - 12 = 18$ $6x = 30$ $x = 5$

Special Cases

No solution (contradiction): $2x + 5 = 2x + 8$ $5 = 8$ ← False! No value of $x$ works.

Infinitely many solutions (identity): $3(x + 2) = 3x + 6$ $3x + 6 = 3x + 6$ ← Always true! Any value of $x$ works.

Equations with Fractions

Multiply every term by the LCD to clear fractions:

$\frac{x}{3} + \frac{x}{4} = 7$

Multiply by 12: $4x + 3x = 84$ $7x = 84$ $x = 12$

Equations with Decimals

Multiply by a power of 10:

$0.3x + 1.5 = 4.2$ Multiply by 10: $3x + 15 = 42$ $3x = 27$, $x = 9$

Always check: Substitute your answer back into the original equation to verify.