Slope-Intercept Form of a Line

Overview

The slope-intercept form of a line is a way to express a linear equation. It allows us to see the slope and y-intercept of the line directly from the equation. The slope-intercept form is given by:

\[ y = mx + b \]

Where:

  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept of the line, or the point where the line crosses the y-axis.

This form is useful for quickly graphing lines and understanding the relationship between x and y values.

Practice Problems

  1. Write the equation of a line with a slope of 3 and a y-intercept of -2.
    Solution

    Using the slope-intercept form \( y = mx + b \):

    \[ y = 3x - 2 \]

  2. Determine the slope and y-intercept for the line represented by the equation \( y = -5x + 7 \).
    Solution

    The slope \( m = -5 \) and the y-intercept \( b = 7 \).

  3. Rewrite the equation \( 2x - y = 4 \) in slope-intercept form and identify the slope and y-intercept.
    Solution

    Rearrange to get \( y = 2x - 4 \):

    The slope \( m = 2 \) and the y-intercept \( b = -4 \).

  4. What is the equation of a line with a slope of -1/2 that passes through the point (0, 3)?
    Solution

    Since it passes through the y-axis at (0, 3), the y-intercept \( b = 3 \). The equation is:

    \[ y = -\frac{1}{2}x + 3 \]

  5. Graph the line \( y = \frac{3}{4}x - 2 \) and identify the slope and y-intercept.
    Solution

    The slope is \( \frac{3}{4} \), and the y-intercept is -2. To graph, start at (0, -2) and move up 3 units and over 4 units to plot additional points.

Additional Practice

Try these additional questions for more practice:

  1. Convert \( 4y - 8x = 12 \) to slope-intercept form and find the slope and y-intercept.
    Solution

    Rearrange to get \( y = 2x + 3 \). Slope \( m = 2 \), y-intercept \( b = 3 \).

  2. If a line has an equation \( y = -3x + 5 \), find two points on the line.
    Solution

    Choose x = 0: \( y = 5 \), so (0, 5). Choose x = 1: \( y = 2 \), so (1, 2).

  3. A line passes through the point (0, -4) and has a slope of 7. Write its equation in slope-intercept form.
    Solution

    The equation is \( y = 7x - 4 \).

→Finding the Equation of a Line