Question

Derive the equation of the line in slope-intercept form. the numbers are (3,0) and (0,-2)

Answer 1

To derive the equation of the line in slope-intercept form (y = mx + b) using the points (3, 0) and (0, -2), we first need to find the slope (m) of the line.

### Step 1: Calculate the slope (m)

The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Using the points (3, 0) and (0, -2), we can designate:

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

Now plug in the values:

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

### Step 2: Use the slope-intercept form

Now that we have the slope \(m = \frac{2}{3}\), we can use one of the points to solve for \(b\) (the y-intercept). We'll use the point (0, -2), which is already in the form needed:

\[
y = mx + b
\]

Plug in \(x = 0\) and \(y = -2\):

\[
-2 = \frac{2}{3}(0) + b \implies -2 = b
\]

### Step 3: Write the final equation

Now that we know \(m\) and \(b\), we can write the equation of the line in slope-intercept form:

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

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

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