Question

Find an equation of a line perpendicular to the given line that contains the given point. Write the equation in slope-intercept form Line 5x-4y=6, point(-4,3)

Answer 1

Step-by-step explanation

From the statement, we must find an equation:

0. of a line perpendicular to the line 5x - 4y = 6,

,

1. that contains the point (x, y) = (-4, 3).

We write the equation of the given line as:

`\begin{gathered} 4y+6=5x, \\ 4y=5x-6, \\ y=(5)/(4)x-(6)/(4)=(5)/(4)x-(3)/(2). \end{gathered}`

We see that the slope of the given line is m₁ = 5/4.

The slope-intercept equation of the line that we must find is:

`y=m_2\cdot x+b_2.`

Where:

• m₂ is the slope,

,

• b₂ is the y-intercept.

1) From condition 1, we know that the line must be perpendicular to the given line. So their slopes must satisfy:

`m_1\cdot m_2=-1\Rightarrow(5)/(4)\cdot m_2=-1.`

Solving for m₂ we get:

`m_2=-(4)/(5).`

2) Replacing the coordinates of the point from condition 2, and the slope m₂ = -4/5 in the equation of the line, we have:

`\begin{gathered} 3=(-(4)/(5))\cdot(-4)+b_2, \\ 3=(16)/(5)+b_2. \end{gathered}`

Solving for b₂ we get:

`b_2=3-(16)/(5)=-(1)/(5).`

So the slope-intercept equation of the perpendicular line is:

`y=-(4)/(5)x-(1)/(5).`Answer
`y=-(4)/(5)x-(1)/(5)`