Question

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

Answer 1

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

1. **Find the slope \(m\):**

The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:

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

Plugging in the coordinates \((x_1, y_1) = (0, -2)\) and \((x_2, y_2) = (1, 3)\):

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

So, the slope \(m\) is 5.

2. **Use the slope-intercept form:**

The slope-intercept form of a line is:

\[
y = mx + b
\]

We already know \(m = 5\), so now we need to find the \(y\)-intercept \(b\).

3. **Find the \(y\)-intercept \(b\):**

Since we know the line passes through the point \((0, -2)\), we can substitute \(x = 0\) and \(y = -2\) into the equation to find \(b\):

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

Simplifying:

\[
-2 = b
\]

So, \(b = -2\).

4. **Write the final equation:**

Substitute \(m\) and \(b\) back into the slope-intercept form:

\[
y = 5x - 2
\]

Therefore, the equation of the line in slope-intercept form is \(y = 5x - 2\).