Question

the point - slope form of the equation of the line that passes through (-9, -2) and (1, 3) is $y - 3=\frac{1}{2}(x - 1)$. what is the slope - intercept form of the equation for this line?
○ $y=\frac{1}{2}x + 2$
○ $y=\frac{1}{2}x - 4$
○ $y=\frac{1}{2}x+\frac{5}{2}$
○ $y=\frac{1}{2}x-\frac{7}{2}$

Explanation:

Step1: Recall slope-intercept form

Slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We need to simplify the given point - slope form \( y - 3=\frac{1}{2}(x - 1) \) to this form.

Step2: Distribute the slope

First, distribute \( \frac{1}{2} \) on the right - hand side of the equation \( y - 3=\frac{1}{2}(x - 1) \). Using the distributive property \( a(b + c)=ab+ac \) (here \( a=\frac{1}{2},b = x,c=- 1 \)), we get \( y - 3=\frac{1}{2}x-\frac{1}{2}\times1 \), which simplifies to \( y - 3=\frac{1}{2}x-\frac{1}{2} \).

Step3: Solve for y

To solve for \( y \), we add 3 to both sides of the equation. We know that \( 3=\frac{6}{2} \), so \( y=\frac{1}{2}x-\frac{1}{2}+3 \). Combining the constant terms: \( -\frac{1}{2}+\frac{6}{2}=\frac{- 1 + 6}{2}=\frac{5}{2} \). So \( y=\frac{1}{2}x+\frac{5}{2} \).

Answer:

\( y=\frac{1}{2}x+\frac{5}{2} \) (which corresponds to the option \( y = \frac{1}{2}x+\frac{5}{2} \))