Question

1. which equation represents a line that is perpendicular to $y = -\frac{1}{5}x - 8$ and passes through the ordered pair $(-5, 7)$?

$\bigcirc\\ y = -\frac{1}{5}x + 7$
$\bigcirc\\ y = 5x + 32$
$\bigcirc\\ y = \frac{1}{5}x + 8$
$\bigcirc\\ y = -5x - 18$

Explanation:

Step1: Find slope of perpendicular line

The slope of the given line \( y = -\frac{1}{5}x - 8 \) is \( m_1 = -\frac{1}{5} \). For perpendicular lines, the product of slopes is \( -1 \), so \( m_1 \times m_2 = -1 \). Solving \( -\frac{1}{5} \times m_2 = -1 \), we get \( m_2 = 5 \).

Step2: Use point - slope form

The point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(-5,7) \) and \( m = 5 \). Substituting, we have \( y - 7 = 5(x + 5) \).

Step3: Simplify the equation

Expand \( y - 7 = 5x + 25 \), then add 7 to both sides: \( y = 5x + 25 + 7 = 5x + 32 \).

Answer:

B. \( y = 5x + 32 \) (assuming the second option is labeled B, if the original options had labels like A, B, C, D with the second option being B for \( y = 5x+32 \))