Determine the equation of a line y = mx + c

To determine the equation of a line in the form \( y = mx + c \), we need to find the slope \( m \) and the y-intercept \( c \). Here's how we calculate them:

1. **Slope \( m \)**:
   The slope of the line is the change in \( y \) divided by the change in \( x \). If you have two points on the line, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope is given by:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

2. **Y-intercept \( c \)**:
   The y-intercept is the point where the line crosses the y-axis. This occurs when \( x = 0 \). To find \( c \), use the equation of the line with a known point \( (x_1, y_1) \) and the calculated slope:
   \[
   y_1 = m x_1 + c
   \]
   Solve for \( c \):
   \[
   c = y_1 - m x_1
   \]

Once you know \( m \) and \( c \), you can write the full equation of the line.

### Example:
Suppose you're given two points: \( (1, 2) \) and \( (3, 6) \).

- The slope is:
  \[
  m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
  \]

- To find the y-intercept, use one of the points, say \( (1, 2) \), in the equation:
  \[
  2 = 2(1) + c \quad \Rightarrow \quad c = 0
  \]

Thus, the equation of the line is:
\[
y = 2x
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you find the slope of a vertical line?
2. What happens when the slope \( m \) is zero?
3. How do you calculate the slope if the points are fractions?
4. Can you determine the equation of a line given only one point and the slope?
5. How can you verify if a point lies on a given line?

### Tip:
To quickly determine the slope, remember it represents how steep the line is—higher values of \( m \) mean a steeper line!

Learn how to determine the equation of a line in the slope-intercept form y = mx + c. This step-by-step guide explains how to calculate the slope and y-intercept, with an example involving points (1, 2) and (3, 6). Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation