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

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{ = }(5x)/(4)\text{ - 4}`

Hence, the gradient of the line is:

`m\text{ = }(5)/(4)`

Step 3:

The gradient that is perpendicular to this, is:

`\begin{gathered} \text{m}_1m_2=\text{ -1} \\ (5)/(4)m_2=\text{ -1} \end{gathered}`

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

`\begin{gathered} m_{2_{}}\text{ = -1 x }(4)/(5)=\text{ }(-4)/(5) \\ m_2=(-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{ = }(-4)/(5) \end{gathered}`

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