Question

Find the equation of the line which passes through the points(-5,8) And is perpendicular to the given line express your answer in slope intercept form simplify your answer

Answer 1

Given: The equation below

`4x+7y=4y-7`

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

Solution

Let us determine the slope of the given equation

`\begin{gathered} 4x+7y=4y-7 \\ 7y-4y=-4x-7 \\ 3y=-4x-7 \\ (3y)/(3)=(-4x)/(3)-(7)/(3) \\ y=-(4)/(3)x-(7)/(3) \end{gathered}`

The slope-intercept form of a linear equation is given as

`\begin{gathered} y=mx+c \\ m=slope \\ c=y-intercept \end{gathered}`

Comparing the slope-intercept form to the given equation

`\begin{gathered} y=-(4)/(3)x-(7)/(3) \\ y=mx+c \\ slope=m=-(4)/(3) \\ c=-(7)/(3) \end{gathered}`

Note: If two lines are perpendicular to each other, the slope of one of the line is equal to the negative inverse of the other

Therefore, the slope of the perpendicular line is as shown below

`\begin{gathered} slope(given-equation)=m \\ slope(perpendicular-line)m_2=-(m)^(-1) \\ So \\ m=-(4)/(3) \\ m_2=-(-(4)/(3))^(-1) \\ m_2=-(-(3)/(4)) \\ m_2=(3)/(4) \end{gathered}`

If the perpendicular line passes through (-5,8), the equation of the line can be derived using the formula below

`\begin{gathered} (y-y_1)/(x-x_1)=slope \\ (x_1,y_1)=(-5,8) \\ slope=(3)/(4) \\ Therefore, \\ (y-8)/(x--5)=(3)/(4) \end{gathered}`
`\begin{gathered} (y-8)/(x+5)=(3)/(4) \\ y-8=(3)/(4)(x+5) \\ y-8=(3)/(4)x+(15)/(4) \\ y=(3)/(4)x+(15)/(4)+8 \end{gathered}`
`\begin{gathered} y=(3)/(4)x+(15+32)/(4) \\ y=(3)/(4)x+(47)/(4) \end{gathered}`

Hence, the equation of the line that passes through the point ( - 5, 8) and perpendicular to the given equation is

`y=(3)/(4)x+(47)/(4)`