Question

write an equation in slope intercept form that passes through the two points shown:
$y = mx + b$
$(-8,-2)$ and $(2,3)$
$y = \frac{1}{2}x + 2$

Explanation:

Step1: Calculate the slope \( m \)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
For points \((-8, -2)\) and \((2, 3)\), \( x_1 = -8 \), \( y_1 = -2 \), \( x_2 = 2 \), \( y_2 = 3 \).
So \( m = \frac{3 - (-2)}{2 - (-8)} = \frac{3 + 2}{2 + 8} = \frac{5}{10} = \frac{1}{2} \).

Step2: Find the y - intercept \( b \)

We know the slope - intercept form is \( y = mx + b \), and we can use one of the points (let's use \((2, 3)\)) and the slope \( m=\frac{1}{2}\) to find \( b \).
Substitute \( x = 2 \), \( y = 3 \), and \( m=\frac{1}{2}\) into \( y = mx + b \):
\( 3=\frac{1}{2}(2)+b \)
Simplify the right - hand side: \( \frac{1}{2}(2)=1 \), so the equation becomes \( 3 = 1 + b \).
Subtract 1 from both sides: \( b=3 - 1=2 \).

Step3: Write the equation in slope - intercept form

Now that we have \( m=\frac{1}{2}\) and \( b = 2 \), the slope - intercept form of the line is \( y=\frac{1}{2}x + 2 \).

Answer:

\( y=\frac{1}{2}x + 2 \)