Solving Linear Equations

Learn to solve one-step, two-step, and multi-step linear equations

What is a Linear Equation?

A linear equation is an equation where the variable appears to the first power only.

Examples: $x + 5 = 12$ , $3x - 7 = 20$ , $2(x + 3) = 5x - 4$

Example: Solve $3x + 4 = 19$

Step 1: Subtract 4 from both sides $3x = 15$

Step 2: Divide by 3 $x = 5$

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

Addition/Subtraction: If $a = b$ , then $a + c = b + c$
Multiplication/Division: If $a = b$ , then $ac = bc$ (and $\frac{a}{c} = \frac{b}{c}$ if $c \neq 0$ )

Solve for $x$ : $2x + 7 = 15$

💡 Show Solution

Step 1: Subtract 7 from both sides $2x + 7 - 7 = 15 - 7$ $2x = 8$

Step 2: Divide both sides by 2 $\frac{2x}{2} = \frac{8}{2}$ $x = 4$

Check: $2(4) + 7 = 8 + 7 = 15$ ✓

Answer: $x = 4$

Solve for $x$ : $5(x - 3) = 2x + 9$

💡 Show Solution

Step 1: Distribute the 5 $5x - 15 = 2x + 9$

Step 2: Subtract $2x$ from both sides $5x - 2x - 15 = 2x - 2x + 9$ $3x - 15 = 9$

Step 3: Add 15 to both sides $3x - 15 + 15 = 9 + 15$ $3x = 24$

Step 4: Divide by 3 $x = 8$

Answer: $x = 8$

Solve for $x$ : $\frac{2x + 3}{4} = \frac{x - 1}{2}$

💡 Show Solution

Step 1: Multiply both sides by 4 (the LCD) $4 \cdot \frac{2x + 3}{4} = 4 \cdot \frac{x - 1}{2}$ $2x + 3 = 2(x - 1)$

Step 2: Distribute $2x + 3 = 2x - 2$

Step 3: Subtract $2x$ from both sides $2x - 2x + 3 = 2x - 2x - 2$ $3 = -2$

This is a contradiction!

Answer: No solution (the equation has no value of $x$ that makes it true)

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