Question

write the equation of a line perpendicular to 5x - 6y = - 7 that passes through the point (- 5,4).
the equation of the line is y = \square.
(simplify your answer. type your answer in slope-intercept form. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find slope of given line

Rewrite \(5x - 6y=-7\) in slope - intercept form \(y = mx + b\) (where \(m\) is the slope).
\(-6y=-5x - 7\), then \(y=\frac{5}{6}x+\frac{7}{6}\). So the slope of the given line \(m_1=\frac{5}{6}\).

Step2: Find slope of perpendicular line

If two lines are perpendicular, the product of their slopes is \(- 1\) (i.e., \(m_1\times m_2=-1\)). Let the slope of the perpendicular line be \(m_2\).
\(\frac{5}{6}\times m_2=-1\), so \(m_2=-\frac{6}{5}\).

Step3: Use point - slope form to find equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-5,4)\) and \(m =-\frac{6}{5}\).
Substitute the values: \(y - 4=-\frac{6}{5}(x + 5)\).

Step4: Convert to slope - intercept form

Expand the right - hand side: \(y - 4=-\frac{6}{5}x-6\).
Add 4 to both sides: \(y=-\frac{6}{5}x-6 + 4\), so \(y=-\frac{6}{5}x-2\).

Answer:

\(y =-\frac{6}{5}x-2\)