Solving Linear Equations

What is a Linear Equation?

A linear equation is an equation where the variable appears to the first power (no exponents or radicals). The graph of a linear equation is always a straight line.

Examples of linear equations:

• $x + 5 = 12$
• $3x - 7 = 20$
• $2(x + 3) = 5x - 4$

Steps to Solve Linear Equations

1. One-Step Equations

For equations like $x + 5 = 12$, perform the inverse operation on both sides:

$x + 5 = 12$ $x + 5 - 5 = 12 - 5$ $x = 7$

2. Two-Step Equations

For equations like $3x + 4 = 19$:

1. Subtract 4 from both sides: $3x + 4 - 4 = 19 - 4$ $3x = 15$

2. Divide both sides by 3: $\frac{3x}{3} = \frac{15}{3}$ $x = 5$

3. Multi-Step Equations

For equations like $2(x + 3) = 5x - 4$:

1. Distribute: $2x + 6 = 5x - 4$
2. Combine like terms: $2x - 5x = -4 - 6$
3. Simplify: $-3x = -10$
4. Solve: $x = \frac{10}{3}$

Key Properties

Addition Property of Equality: If $a = b$, then $a + c = b + c$

Subtraction Property of Equality: If $a = b$, then $a - c = b - c$

Multiplication Property of Equality: If $a = b$, then $ac = bc$

Division Property of Equality: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$

Common Mistakes to Avoid

• Forgetting to perform the same operation on both sides
• Sign errors when dealing with negative numbers
• Not combining like terms before solving
• Dividing by zero