Question

given the following coordinates, which of the following equations will not lead you to creating the point - slope form of the equation:
(-7, 13) and (-12, -8).
\\(y + (-13) = \frac{21}{5}(x - (-7))\\)
\\(y - 8 = \frac{21}{5}(x - (-12))\\)
\\(y - 13 = \frac{21}{5}(x + 7)\\)
\\(y + 8 = \frac{21}{5}(x + 12)\\)

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((-7, 13)\) and \((-12, -8)\), we have \( x_1=-7,y_1 = 13,x_2=-12,y_2=-8 \). So \( m=\frac{-8 - 13}{-12-(-7)}=\frac{-21}{-5}=\frac{21}{5} \).

Step2: Recall point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \((x_1,y_1)\) is a point on the line and \( m \) is the slope.

Analyze Option 1: \( y+(- 13)=\frac{21}{5}(x-(-7)) \)

This can be rewritten as \( y - 13=\frac{21}{5}(x + 7) \). Using the point \((-7,13)\) (\(x_1=-7,y_1 = 13\)) and slope \( m = \frac{21}{5}\), this is a valid point - slope form.

Analyze Option 2: \( y - 8=\frac{21}{5}(x-(-12)) \)

The point \((-12,-8)\) has \( y_1=-8 \), not \( 8 \). So \( y - 8=\frac{21}{5}(x + 12) \) uses an incorrect \( y \) - coordinate for the point \((-12,-8)\).

Analyze Option 3: \( y - 13=\frac{21}{5}(x + 7) \)

Using the point \((-7,13)\) (\(x_1=-7,y_1 = 13\)) and slope \( m=\frac{21}{5}\), since \( x-(-7)=x + 7 \) and \( y - 13\) is correct, this is a valid point - slope form.

Analyze Option 4: \( y + 8=\frac{21}{5}(x + 12) \)

Using the point \((-12,-8)\) (\(x_1=-12,y_1=-8\)), we have \( y-(-8)=y + 8 \) and \( x-(-12)=x + 12 \) with slope \( m=\frac{21}{5}\), so this is a valid point - slope form.

Answer:

B. \( y - 8=\frac{21}{5}(x-(-12)) \)