Question

1. write the equation of the line that goes through the point (-2,8) and is perpendicular to the line $y + 3 = -\frac{2}{7}(x - 4)$. determine each form.

point - slope form
slope - intercept form
parallel to $x = 12$ and passes through (-7,

Explanation:

Step1: Find the slope of the given line

The given line is \( y + 3 = -\frac{2}{7}(x - 4) \). In point - slope form \( y - y_1=m(x - x_1) \), the slope \( m \) of the given line is \( -\frac{2}{7} \). If two lines are perpendicular, the slope of the perpendicular line \( m_{\perp} \) is the negative reciprocal of the original slope. So, \( m_{\perp}=\frac{7}{2} \) (since the negative reciprocal of \( -\frac{2}{7} \) is \( \frac{7}{2} \)).

Step2: Use point - slope form to write the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-2,8) \) and \( m = \frac{7}{2} \). Substituting these values into the formula, we get \( y - 8=\frac{7}{2}(x + 2) \) (because \( x_1=-2 \), so \( x-(-2)=x + 2 \)).

Step3: Convert to slope - intercept form (\( y=mx + b \))

Start with \( y - 8=\frac{7}{2}(x + 2) \).
First, distribute the \( \frac{7}{2} \) on the right - hand side: \( y - 8=\frac{7}{2}x+7 \).
Then, add 8 to both sides: \( y=\frac{7}{2}x+7 + 8 \).
Simplify the constants: \( y=\frac{7}{2}x+15 \).

Answer:

• Point - Slope Form: \( y - 8=\frac{7}{2}(x + 2) \)
• Slope - Intercept Form: \( y=\frac{7}{2}x+15 \)