What is a Linear Equation in One Variable? A linear equation in one variable is an equation that can be written in the form \(ax + b = 0\), where \(a\) and \(b\) are constants, \(a \neq 0\), and \(x\) is the variable. The highest power of the variable is 1.
Examples:
- \(2x + 3 = 7\) (linear)
- \(5x - 4 = 6x + 2\) (linear)
- \(x^2 + 3 = 5\) (not linear – power is 2)
What does it mean to "solve" an equation? Solving an equation means finding the value(s) of the variable that make the equation true. This value is called the solution or root of the equation.
Rules for Solving Linear Equations:
| Rule | Explanation | Example |
|---|---|---|
| Addition/Subtraction Property | Add or subtract same number from both sides | \(x - 5 = 10 \Rightarrow x = 15\) |
| Multiplication/Division Property | Multiply or divide both sides by same non-zero number | \(3x = 12 \Rightarrow x = 4\) |
| Combining Like Terms | Simplify each side before solving | \(2x + 3x = 5 \Rightarrow 5x = 5\) |
| Transposition | Move terms from one side to the other (change sign) | \(x + 4 = 10 \Rightarrow x = 10 - 4\) |
Step-by-Step Solving Process:
- Simplify both sides (combine like terms, remove brackets)
- Collect variable terms on one side, constant terms on the other
- Isolate the variable using multiplication/division
- Check your answer by substituting back