Question

Derive the equation of the line in slope-intercept form.

The line has two points (1, 8) and (0, 4)

Answer 1

To derive the equation of the line in slope-intercept form ( \( y = mx + b \) ), we first need to find the slope ( \( m \) ) of the line that passes through the points \( (1, 8) \) and \( (0, 4) \).

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

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

Let's assign the points:
- \( (x_1, y_1) = (1, 8) \)
- \( (x_2, y_2) = (0, 4) \)

Now, substituting the coordinates into the slope formula:

\[
m = \frac{4 - 8}{0 - 1} = \frac{-4}{-1} = 4
\]

So, the slope \( m = 4 \).

Now that we have the slope, we can use one of the points to find the y-intercept ( \( b \) ). The y-intercept is the value of \( y \) when \( x = 0 \). We can use the point \( (0, 4) \), where we already know that when \( x = 0 \), \( y = 4 \). Thus, \( b = 4 \).

Now we can write the equation in slope-intercept form:

\[
y = mx + b
\]

Substituting the values of \( m \) and \( b \):

\[
y = 4x + 4
\]

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

\[
\boxed{y = 4x + 4}
\]