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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?

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

B. \(y=\frac{1}{2} x-4\)

C. \(y=\frac{1}{2} x+\frac{5}{2}\)

D. [tex]\(y=\frac{1}{2} x-\frac{7}{2}\)[/tex]



Answer :

GinnyAnswer
Let's start by converting the given equation from point-slope form to slope-intercept form. The point-slope form of the equation given is:

[tex]\[ y - 3 = \frac{1}{2}(x - 1) \][/tex]

To convert this to the slope-intercept form \( y = mx + b \), we need to:

1. Distribute the slope on the right-hand side.
2. Isolate \( y \) on the left-hand side.

First, let's distribute \(\frac{1}{2}\) to both \(x\) and \(-1\):

[tex]\[ y - 3 = \frac{1}{2} x - \frac{1}{2} \][/tex]

Next, we need to isolate \( y \). To do this, we add 3 to both sides of the equation:

[tex]\[ y = \frac{1}{2} x - \frac{1}{2} + 3 \][/tex]

Now, let's combine the constants on the right-hand side:

[tex]\[ y = \frac{1}{2} x - \frac{1}{2} + \frac{6}{2} \][/tex]

Since \( 3 \) can be written as \( \frac{6}{2} \), we add the constants:

[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]

So, the slope-intercept form of the equation is:

[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]

Among the given choices, this matches the third option:

[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]

Therefore, the slope-intercept form of the equation for this line is:

[tex]\[ y = \frac{1}{2} x + \frac{5}{2} \][/tex]

So, the correct choice is:
[tex]\[ \boxed{y = \frac{1}{2}x + \frac{5}{2}} \][/tex]

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