Understanding Linear Equations in One Variable: A Study Guide

Solving Linear Equations in One Variable

Linear equations can be written in the form ax + b = c, where x is the variable you're solving for, and a, b, and c are just numbers. Your goal is to find the value of x that makes the equation true.

The number of solutions depends on the value of a. When a isn't zero, you'll get exactly one solution. But if a equals zero, things get interesting - you'll either have no solution (when b ≠ c) or infinitely many solutions $when b = c$, which we call an identity.

💡 When solving equations, remember you're using the addition and multiplication properties of equality - whatever you do to one side, you must do to the other!

Let's see this in action with examples:

Example 1: To solve 8x - 12 = 3x + 7, first group like terms by subtracting 3x from both sides to get 5x - 12 = 7. Add 12 to both sides: 5x = 19. Finally, divide by 5 to find x = 3.8.

Example 2: For 4x - 2$x + 6$ = 3$2x + 7$ + 5, start by distributing: 4x - 2x - 12 = 6x + 21 + 5. Simplify to 2x - 12 = 6x + 26, subtract 2x from both sides, then subtract 26 from both sides. You'll get -38 = 4x, so x = -9.5.