Question

2.5 hw part 3: perpendicular slope in slope-intercept form
for the equation of the line described below:
through (5, 2), perpendicular to $y = -\frac{5}{4}x - 5$
first, write the equation of the perpendicular line in point - slope form: $y\square=\square x\square$
next, write the equation in slope - intercept form: $y = \square x\square$
options: $\frac{5}{4}$, $-\frac{5}{4}$, $\frac{4}{5}$, $-\frac{4}{5}$, $+2$, $-2$, $+5$, $-5$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = -\frac{5}{4}x - 5 \) is \( m_1 = -\frac{5}{4} \). The slope of a line perpendicular to it, \( m_2 \), satisfies \( m_1 \times m_2 = -1 \). So, \( -\frac{5}{4} \times m_2 = -1 \), which gives \( m_2 = \frac{4}{5} \).

Step2: Write the point - slope form

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(5,2) \) and \( m=\frac{4}{5} \). Substituting these values, we get \( y - 2=\frac{4}{5}(x - 5) \).

Step3: Convert to slope - intercept form

Start with \( y - 2=\frac{4}{5}(x - 5) \). Expand the right - hand side: \( y - 2=\frac{4}{5}x-4 \). Then add 2 to both sides: \( y=\frac{4}{5}x-4 + 2=\frac{4}{5}x-2 \).

Answer:

• Point - slope form: \( y - 2=\frac{4}{5}(x - 5) \) (So the blanks are filled as \( y-2=\frac{4}{5}(x - 5) \), i.e., the first blank: \(-2\), the second blank: \(\frac{4}{5}\), the third blank: \(-5\))
• Slope - intercept form: \( y=\frac{4}{5}x-2 \) (So the blanks are filled as \( y = \frac{4}{5}x-2 \), i.e., the first blank: \(\frac{4}{5}\), the second blank: \(-2\))