Equation of a Line

What is the Equation of a Line?

The equation of a line describes all the points that lie on the line in a Cartesian plane. The most common forms are:

Slope-Intercept Form:

$$y = mx + c$$

where:

• $m$ is the slope of the line.
• $c$ is the $y-intercept$ (where the line crosses the $y-axis$).

Point-Slope Form:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a known point on the line.

General Form:

$$ax + by + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$ or $b \neq 0$

Vertical Line Equation:

$$x = k$$

where $k$ is a constant.

Horizontal Line Equation:

$$y = k$$

where $k$ is a constant.

Finding the Equation of a Line

To determine the equation of a line, you need either:

1. The slope ($m$) and the $y-intercept$ ($c$).
2. Two points on the line, from which the slope can be calculated.

Slope Formula

The slope mm between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}, \quad x_1 \neq x_2$$

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Worked Examples

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Summary

• Equation forms:
  1. Slope-Intercept: $y = mx + c$
  2. Point-Slope: $y - y_1 = m(x - x_1)$
  3. General Form: $ax + by + c = 0$
  4. Vertical Line: $x = k$
  5. Horizontal Line: $y = k$
• Key Formula: Slope, $m = \frac{y_2 - y_1}{x_2 - x_1}$
• To find the equation, use the slope and one point or two points on the line.
• Practice finding equations for different line forms to solidify your understanding.