Full Prep Academy

Overview

To find the equation of a line, we often use the slope-intercept form:

\[ y = mx + b \]

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.

If we know the slope and a point on the line, or two points on the line, we can find the equation by following these steps:

1. Calculate the slope \( m \) if not given, using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
2. Use the slope-intercept form \( y = mx + b \) or the point-slope form \( y - y_1 = m(x - x_1) \) with a known point \((x_1, y_1)\) to solve for \( b \), the y-intercept.
3. Write the final equation in slope-intercept form.

Practice Problems

1. Find the equation of a line with a slope of 4 that passes through the point (1, 2).
   Solution

   Using \( y - y_1 = m(x - x_1) \):

   \[ y - 2 = 4(x - 1) \Rightarrow y = 4x - 2 \]

2. Find the equation of the line that passes through the points (3, 7) and (5, 11).
   Solution

   Calculate the slope: \[ m = \frac{11 - 7}{5 - 3} = 2 \]

   Use \( y - 7 = 2(x - 3) \):

   \[ y = 2x + 1 \]

3. A line passes through the point (0, -3) and has a slope of -1. Find its equation.
   Solution

   Since it passes through (0, -3), the y-intercept \( b = -3 \):

   \[ y = -x - 3 \]

4. Find the equation of a line that passes through (-2, 5) and (2, -3).
   Solution

   Calculate the slope:

   \[ m = \frac{-3 - 5}{2 - (-2)} = -2 \]

   Use \( y - 5 = -2(x + 2) \):

   \[ y = -2x + 1 \]

5. Write the equation of a line with a y-intercept of 4 and passing through the point (3, -2).
   Solution

   Using \( y = mx + 4 \), solve for \( m \) using (3, -2):

   \[ -2 = m(3) + 4 \Rightarrow m = -2 \]

   The equation is:

   \[ y = -2x + 4 \]

Additional Practice

Try these questions for more practice:

1. Find the equation of a line passing through (1, 3) and with a y-intercept of 6.
2. A line has a slope of 3/4 and passes through the origin. What is its equation?
3. Write the equation of a line passing through (6, 2) and (8, -1).
4. Determine the equation of a line with slope -5 that crosses the y-axis at (0, 2).