Question

find the equation of the line perpendicular to y = (5/6)x + 7, that passes through (5, -2). y = -?/ x +

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{5}{6}x + 7 \) is \( m_1=\frac{5}{6} \). For two perpendicular lines, the product of their slopes is -1, i.e., \( m_1\times m_2=-1 \). So, \( \frac{5}{6}\times m_2 = - 1 \), solving for \( m_2 \) gives \( m_2=-\frac{6}{5} \). So the slope of the perpendicular line is \( -\frac{6}{5} \), meaning the numerator is 6 and the denominator is 5.

Step2: Use point - slope form to find the y - intercept

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(5,-2) \) and \( m = -\frac{6}{5} \). Substitute these values into the formula:
\( y-(-2)=-\frac{6}{5}(x - 5) \)
Simplify the left - hand side: \( y + 2=-\frac{6}{5}(x - 5) \)
Distribute the \( -\frac{6}{5} \) on the right - hand side: \( y+2=-\frac{6}{5}x+6 \)
Subtract 2 from both sides to solve for \( y \): \( y=-\frac{6}{5}x + 6-2 \), so \( y=-\frac{6}{5}x+4 \)

Answer:

The equation of the line is \( y = -\frac{6}{5}x + 4 \), so the numerator of the fraction is 6, the denominator is 5, and the constant term is 4.