Introduction

In mathematics, linear equations are equations that represent straight lines on a graph. They are in the form $y = mx + c$, where $m$ is the slope of the line and $c$ is the y-intercept.

Definition of a Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the first power.

Example: Solve the linear equation $2x + 4 = 10$ for $x$. Solution:
$2x + 4 = 10$
$2x = 10 - 4$
$2x = 6$
$x = \frac{6}{2}$
$x = 3$

Finding the Slope

The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

Example: Find the slope of the line passing through the points $(2, 3)$ and $(5, 9)$. Solution:
Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$
$m = \frac{9 - 3}{5 - 2}$
$m = \frac{6}{3}$
$m = 2$

Solving Simultaneous Linear Equations

Simultaneous linear equations are a set of two or more linear equations containing the same variables.

Example: Solve the simultaneous equations:
$3x + 2y = 8$
$5x - y = 2$ Solution:
Multiply the second equation by 2 to eliminate $y$:
$3x + 2y = 8$
$10x - 2y = 4$
Add the equations:
$13x = 12$
$x = \frac{12}{13}$
Substitute $x$ back into one of the original equations to find $y$.

Common Mistakes

• Misidentifying the slope-intercept form of a linear equation.
• Incorrectly applying the slope formula.
• Making errors in the process of solving simultaneous linear equations.

Key Points

• Linear equations represent straight lines on a graph.
• The slope of a line indicates its steepness.
• Simultaneous linear equations involve solving multiple equations with the same variables simultaneously.

Practice Questions

1. Solve the linear equation $4x - 7 = 5$.
2. Find the slope of the line passing through the points $(1, 4)$ and $(3, 10)$.
3. Solve the simultaneous equations:
   $2x + 3y = 5$
   $4x - y = 7$