Question

line bn is represented by the equation $y = \frac{3}{4}x + \frac{7}{4}$. determine the equation, in slope - intercept form, of the line hz that is perpendicular to line bn and passes through the point $h(-6, 7)$. $y = -\frac{4}{3}x - 1$ (there is also a table with columns slope of line bn, slope of line hz, point - slope form of line hz and a graph on the right side showing points b and h and two lines.)

Explanation:

Step1: Find slope of BN

Line BN: \( y = \frac{3}{4}x + \frac{7}{4} \), so \( m_1 = \frac{3}{4} \).

Step2: Find slope of HZ (perpendicular)

For perpendicular lines, \( m_1 \cdot m_2 = -1 \). So \( \frac{3}{4} \cdot m_2 = -1 \), \( m_2 = -\frac{4}{3} \).

Step3: Point - Slope Form

Point \( H(-6, 7) \), use \( y - y_1 = m(x - x_1) \). Substitute \( m = -\frac{4}{3} \), \( x_1 = -6 \), \( y_1 = 7 \):
\( y - 7 = -\frac{4}{3}(x - (-6)) \)
\( y - 7 = -\frac{4}{3}(x + 6) \)

Step4: Convert to Slope - Intercept

Expand: \( y - 7 = -\frac{4}{3}x - 8 \)
Add 7: \( y = -\frac{4}{3}x - 1 \)

Now fill the table:

• Slope of Line BN (\( m_1 \)): \( \frac{3}{4} \)
• Slope of Line HZ (\( m_2 \)): \( -\frac{4}{3} \)
• Point - Slope Form of Line HZ: \( y - 7 = -\frac{4}{3}(x + 6) \)

Answer:

Slope of Line BN	Slope of Line HZ	Point - Slope Form of Line HZ

And the equation of line HZ in slope - intercept form is \( y = -\frac{4}{3}x - 1 \)