Question

Use slopes to determine if the lines -x-5y=7 and the lines 5x-y= -9 are perpendicular?

Answer 1

Given:

`\begin{gathered} Line\text{ 1:} \\ -x-5y=7 \\ \\ Line\text{ 2:} \\ 5x-y=-9 \end{gathered}`

Rewriting the equations of the lines given in the slope-intercept form to get the slopes of the lines;

`\begin{gathered} y=mx+b \\ where: \\ m\text{ is the slope} \\ b\text{ is the y-intercept} \\ \\ Then,\text{ the lines can be rewritten as;} \\ Line\text{ 1:} \\ -x-5y=7 \\ -5y=7+x \\ -5y=x+7 \\ y=(x+7)/(-5) \\ Splitting\text{ the terms;} \\ y=-(1)/(5)x-(7)/(5) \\ Hence,\text{ the slope of line 1 is;} \\ m_1=-(1)/(5) \end{gathered}`

Also, for line 2

`\begin{gathered} Rewriting\text{ line 2;} \\ Line\text{ 2:} \\ 5x-y=-9 \\ 5x+9=y \\ y=5x+9 \\ Then,\text{ the slope of line 2 is;} \\ m_2=5 \end{gathered}`

Two lines are perpendicular when the product of their slopes is -1.

Hence,

`If\text{ }m_1m_2=-1,\text{ then the lines are perpendicular}`

Multiplying the slopes of the two lines gotten;

`\begin{gathered} m_1m_2=-(1)/(5)*5 \\ m_1m_2=-1 \end{gathered}`

Since the product of their slopes is -1, then the two lines are perpendicular.