Question

2.5 hw part 3: perpendicular slope in slope-intercept form
for the equation of the line described below:
through $(-3, -5)$, perpendicular to $y = -x - 5$
first, write the equation of the perpendicular line in point - slope form: $y\boxed{\quad}=\boxed{\quad}(x\boxed{\quad})$
next, write the equation in slope - intercept form: $y = \boxed{\quad}x\boxed{\quad}$
options: $\pm1$, $\pm2$, $\pm3$, $\pm5$

Explanation:

Step1: Find the slope of the perpendicular line

The given line is \( y = -x - 5 \), so its slope \( m_1=-1 \). For perpendicular lines, the product of slopes is \( -1 \), so the slope of the perpendicular line \( m_2 \) satisfies \( m_1\times m_2=-1 \). Substituting \( m_1 = - 1 \), we get \( -1\times m_2=-1 \), so \( m_2 = 1 \).

Step2: Write the point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-3,-5) \) and \( m = 1 \). Substituting these values, we have \( y-(-5)=1\times(x - (-3)) \), which simplifies to \( y + 5=1(x + 3) \).

Step3: Convert to slope - intercept form

Start with \( y + 5=x + 3 \). Subtract 5 from both sides: \( y=x+3 - 5 \), so \( y=x - 2 \).

Answer:

• Point - slope form: \( y+5 = 1(x + 3) \)
• Slope - intercept form: \( y = 1x-2 \)