Question

what is the equation of a line that is perpendicular to the line of the graph and passes through the point 4 , 6

Answer 1

If two lines are perpendicular, the product of their slopes must be equal to -1.

Find the slope of the line on the graph. Then, find the slope of a line perpendicular to it. Finally, use the slope-point form of the equation of a line to find the equation of a line that is perpendicular to the line on the graph and passes through the point (4,6).

To find the slope of the line on the graph, identify the coordinates of two points on the line and use the slope formula:

`m=(y_2-y_1)/(x_2-x_1)`

Let m_1 be the slope of the line of the graph. Notice that the points (-4,7) and (-3,5) belong to that line. Use those coordinates to find the slope m_1:

`\begin{gathered} m_1=((7)-(5))/((-4)-(-3)) \\ =(7-5)/(-4+3) \\ =(2)/(-1) \\ =-2 \end{gathered}`

Let m_2 be the slope of a line perpendicular to the line of the graph. Then:

`m_1m_2=-1`

Solve for m_2:

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

Find the value of m_2, provided that m_1=-2:

`\begin{gathered} m_2=-(1)/(-2) \\ \therefore m_2=(1)/(2) \end{gathered}`

The slope-point form of the equation of a line, is:

`y=m(x-x_0)+y_0`

Therefore, the equation of a line with slope 1/2 that passes through the point (4,6) is:

`y=(1)/(2)(x-4)+6`

Which is the equation that we were looking for.