Equations of Lines and Their Graphs

Equations of Lines in the Coordinate Plane

Finding the Equation of a Line in Slope-Intercept Form

The slope-intercept form of a line is a common way to express the equation of a straight line. It is written as:

• General form: $y = mx + b$

• Where m is the slope and b is the y-intercept.

To find the equation of a line passing through two points, follow these steps:

1. Find the slope (m): Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.

2. Substitute one point and the slope into the slope-intercept form to solve for b.

3. Write the final equation.

Example: Find the equation of the line passing through $(-3, -3)$ and $(6, 3)$.

• Step 1: Find the slope: $m = \frac{3 - (-3)}{6 - (-3)} = \frac{6}{9} = \frac{2}{3}$

• Step 2: Substitute one point (e.g., $(-3, -3)$) into $y = mx + b$: $-3 = \frac{2}{3}(-3) + b$ $-3 = -2 + b$ $b = -1$

• Step 3: Write the equation: $y = \frac{2}{3}x - 1$

Point-Slope Form of a Line

The point-slope form is another way to write the equation of a line, especially useful when you know a point on the line and the slope. The form is:

• General form: $y - y_1 = m(x - x_1)$

• Where $(x_1, y_1)$ is a point on the line and m is the slope.

Example: Using the same points $(-3, -3)$ and $(6, 3)$, and the previously found slope $m = \frac{2}{3}$:

• Using $(-3, -3)$: $y - (-3) = \frac{2}{3}(x - (-3))$ $y + 3 = \frac{2}{3}(x + 3)$

Graphing Linear Equations and Intercepts

Graphing a Line Given Slope and Y-Intercept

To graph a line when given the slope and y-intercept, use the slope-intercept form $y = mx + b$.

• Start at the y-intercept (b) on the y-axis.

• Use the slope (m) to find another point: Rise over run (for $m = \frac{1}{3}$, go up 1 unit and right 3 units).

• Draw the line through these points.

Example: For a line with slope $\frac{1}{3}$ and y-intercept $-5$:

• Equation: $y = \frac{1}{3}x - 5$

• Start at (0, -5) on the y-axis.

• From (0, -5), move up 1 and right 3 to (3, -4).

Finding the X-Intercept of a Line

The x-intercept is the point where the line crosses the x-axis (i.e., where $y = 0$).

• Set $y = 0$ in the equation and solve for $x$.

Example: For $y = \frac{1}{3}x - 5$:

• Set $y = 0$:

• $0 = \frac{1}{3}x - 5$

• $\frac{1}{3}x = 5$

• $x = 15$

• The x-intercept is at $(15, 0)$.

Summary Table: Forms of Linear Equations

Form	Equation	When to Use
Slope-Intercept	$y = mx + b$	When you know the slope and y-intercept
Point-Slope	$y - y_1 = m(x - x_1)$	When you know the slope and a point
Standard	$Ax + By = C$	General form, useful for certain algebraic manipulations