Question

Write the equation of a line that is perpendicular to the line y - 2 = 4(x – 1)that goes through point (-4,5).

Answer 1

If two lines are perpendicular, then their slopes are opposite reciprocals.

This means that if you consider the lines:

`y_1=m_1x_1+b_1`
`y_2=m_2x_2+b_2`

That are perpendicular, the relationship between their slopes is the following:

`m_2=-(1)/(m_1)`

To determine the equation of a line perpendicular to y-2=4(x-1), the first step is to determine the value of its slope. This equation is given in the point-slope form which has the following structure:

`y-y_1=m(x-x_1)`

Where

m represents the slope of the line

(x₁,y₁) represent the coordinates of one point of the line.

On the given equation, the slope is the coefficient that multiplies the parentheses term, m=4

We know that the slope of a line perpendicular to the given line will be the inverse opposite of m=4, then the slope of the perpendicular line will be:

`m=-(1)/(4)`

Using the coordinates of the given point (-4,5), the slope m=-1/4, and the point-slope form, you can determine the equation as follows:

`\begin{gathered} y-y_1=m(x-x_1) \\ y-5=-(1)/(4)(x-(-4)) \\ y-5=-(1)/(4)(x+4) \end{gathered}`

So, the equation of the line, that is perpendicular to y-2=4(x-1) and passes through the point (-4,5) is

`y-5=-(1)/(4)(x+4)`