Linear Equations
A linear equation is an equation that forms a straight line when plotted on a coordinate plane. In its standard form, a linear equation with one variable can be written as:
$$ax + b = 0$$
Where $a$ and $b$ are constants, and $a \neq 0$.
The Solution Process
To solve a linear equation:
- Simplify both sides by combining like terms
- Use the addition property of equality to isolate variable terms on one side
- Use the multiplication property of equality to isolate the variable
- Check your solution by substituting it back into the original equation
Example: Solving a Simple Equation
Let’s solve the equation: $3x - 7 = 8$
Add 7 to both sides: $3x - 7 + 7 = 8 + 7$ $3x = 15$
Divide both sides by 3: $\frac{3x}{3} = \frac{15}{3}$ $x = 5$
Check: $3(5) - 7 = 15 - 7 = 8$ ✓
Solving Equations with Variables on Both Sides
Let’s solve: $4x - 3 = 2x + 5$
Subtract $2x$ from both sides: $4x - 2x - 3 = 2x - 2x + 5$ $2x - 3 = 5$
Add 3 to both sides: $2x - 3 + 3 = 5 + 3$ $2x = 8$
Divide both sides by 2: $\frac{2x}{2} = \frac{8}{2}$ $x = 4$
Check: $4(4) - 3 = 16 - 3 = 13$ and $2(4) + 5 = 8 + 5 = 13$ ✓
Solutions to Chapter 1 Practice Problems
- $4x + 3y - 2x + y = 2x + 4y$
- $2(3x - 4) + 5x = 6x - 8 + 5x = 11x - 8$
- $\frac{x^2 - 4}{x - 2} = \frac{(x+2)(x-2)}{x-2} = x + 2$ (where $x \neq 2$)
Practice Problems
Try solving these equations:
- $5x + 3 = 18$
- $7x - 4 = 3x + 12$
- $2(x + 3) = 3(x - 1) + 4$
Solutions will be discussed in the next chapter.