Q24. - Algebra 1

Question

Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation.

$\left(-4,-5\right);-4x+5y=-6$

Step-by-Step Solution

Answer

The equation in the slope-intercept form for the line that passes through the point $\left(-4,-5\right)$ and is perpendicular to the given equation is $y=-\frac{5}{4}x-10$.

Step 1. State the concept of slope-intercept of an equation and slope of parallel and perpendicular lines.

The slope-intercept form of a linear equation is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

Parallel lines: Slopes $m_1$ and $m_2$ of the parallel lines are same i.e., $m_1=m_2$.

Perpendicular lines: The slopes $m_1$ and $m_2$ of the nonvertical perpendicular lines are opposite reciprocals i.e., $m_1{m}_2=-1$. 

Step 2. Calculate the slope.

Calculate the slope of the first equation $-4x+5y=-6$. Convert the given equation in the standard slope and intercept form of the equation $y=mx+b$.

$$\begin{gathered} -4x+5y=-6 \\ 5y=4x-6 \\ y=\frac{4}{5}x-\frac{6}{5} \end{gathered}$$

Compared from the standard form of the equation $y=mx+b$, the slope of the equation is:

 $m=\frac{4}{5}$

Calculate the slope of the line which is perpendicular to the given line $-4x+5y=-6$.

$$\begin{gathered} m_1{m}_2=-1 \\ \left(\frac{4}{5}\right)m_2=-1 \\ m_2=-\frac{5}{4} \end{gathered}$$

Step 3. Write a new equation in slope-intercept form.

To find the equation in slope-intercept form for the line that passes through the point $\left(-4,-5\right)$ and is perpendicular to the equation $-4x+5y=-6$, substitute the value of slope and point in the standard form of slope- point form of the equation i.e., $y-y_1=m\left(x-x_1\right)$ where in $x_1=-4,y_1=-5$. 

 $$\begin{gathered} y-y_1=m\left(x-x_1\right) \\ y-\left(-5\right)=-\frac{5}{4}\left(x-\left(-4\right)\right) \\ y+5=-\frac{5}{4}x-5 \\ y=-\frac{5}{4}x-10 \end{gathered}$$

Therefore, the equation in the slope-intercept form for the line that passes through the point $\left(-4,-5\right)$ and is perpendicular to the given equation is $y=-\frac{5}{4}x-10$.