Question

Derive the equation of the line in slope-intercept form for points (1,3) and (0,-2)

Answer 1

To derive the equation of the line in slope-intercept form (y = mx + b) based on the points (1, 3) and (0, -2), we will follow these steps:

1. **Determine the slope (m)** of the line. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use the points (1, 3) and (0, -2):

\((x_1, y_1) = (1, 3)\) and \((x_2, y_2) = (0, -2)\)

Substitute the values into the slope formula:

\[
m = \frac{-2 - 3}{0 - 1} = \frac{-5}{-1} = 5
\]

2. **Substitute the slope (m) and one of the points into the point-slope form of the equation.** We can use point (0, -2) to find \(b\):

The slope-intercept form of an equation is:

\[
y = mx + b
\]

Plug in the slope \(m = 5\) and the point (0, -2):

\(-2 = 5(0) + b\)

Simplifying this, we get:

\(-2 = 0 + b\)

Thus:

\[
b = -2
\]

3. **Write the final equation** in slope-intercept form:

\[
y = 5x - 2
\]

Thus, the equation of the line in slope-intercept form is:

\[
y = 5x - 2
\]