Question

What is an equation of the line that passes through the point (5, -8) and isperpendicular to the line 5x -4y = 16?

Answer 1

SOLUTION:

Step 1:

In this question, we are given the following:

Step 2:

Given,

$\begin{gathered} \text{5 x - 4 y = 16} \\ \text{comparing with y = mx + c} \\ -4y\text{ = -5x + 16} \end{gathered}$

Divide both sides by -4, we have that:

$y\text{ = }\frac{5x}{4}\text{ - 4}$

Hence, the gradient of the line is:

$m\text{ = }\frac{5}{4}$

Step 3:

The gradient that is perpendicular to this, is:

$\begin{gathered} \text{m}_1m_2=\text{ -1} \\ \frac{5}{4}m_2=\text{ -1} \end{gathered}$

Divide both sides by 5/4, we have that:

$\begin{gathered} m_{2_{}}\text{ = -1 x }\frac{4}{5}=\text{ }\frac{-4}{5} \\ m_2=\frac{-4}{5} \end{gathered}$

Step 4:

The equation of the line that passes through the point (5, -8) and is perpendicular to -4/5 is:

$\begin{gathered} y-y_1=m_{2\text{ }}(x-x_1) \\ where(x_1,y_1)=(5,-8) \\ and_{} \\ m_{\text{ 2}}\text{ = }\frac{-4}{5} \end{gathered}$

$\begin{gathered} y\text{ -( - 8 ) = }\frac{-\text{ 4}}{5}\text{ ( x - }5) \\ y\text{ + 8 =}\frac{-4}{5}(\text{ x -5)} \end{gathered}$

Multiply through by 5, we have that:

$\begin{gathered} 5y\text{ + 40 = - 4 ( x -5)} \\ 5y\text{ + 40 =-4x +20} \\ 5y\text{ + 4x = 20 -40} \\ 5\text{ y + 4x = -20} \end{gathered}$

CONCLUSION:

The equation of the line that passes through the point (5, -8) and is

perpendicular to the line 5x -4y = 16 is:

$5y\text{ +4x = -20}$