Question

Write an equation of the line that is perpendicular to y – x = 5 and passes through the point (8, 3). Which one of the following matches the correct equation for this perpendicular line?Group of answer choicesy = 3x − 8y = 5 − xy = 11 − xy = x – 8

Answer 1

Given:

`\begin{gathered} y-x=5 \\ \\ \text{Through the point (8,3)} \end{gathered}`

The first line given is y - x = 5

`\begin{gathered} Hence, \\ y=x+5 \end{gathered}`

To get the slope of line 1, we compare the equation with the general equation of a straight line.

`\begin{gathered} y=mx+b \\ \text{where m is the slope} \end{gathered}`

Thus,

`\begin{gathered} y=mx+b \\ y=x+5 \\ \\ \text{Comparing both equations,} \\ m=1 \end{gathered}`

The slope of line 1 is 1.

Since line 2 is perpendicular to line 1, then their slopes are negative reciprocals of one another.

This means the product of their slopes is -1.

`m_1m_2=-1`

Hence, the slope of line 2 is gotten by;

`\begin{gathered} m_1m_2=-1 \\ 1* m_2=-1 \\ m_2=-(1)/(1) \\ m_2=-1 \\ \\ \text{The slope of line 2 is -1} \end{gathered}`

Hence, the equation of the perpendicular line through the point (8,3) will be;

`\begin{gathered} (y-y_1)/(x-x_1)=m \\ \text{where;} \\ x_1=8 \\ y_1=3 \\ m=-1 \\ \\ (y-3)/(x-8)=-1 \\ \text{Cross multiplying,} \\ y-3=-1(x-8) \\ y-3=-x+8 \\ y=-x+8+3 \\ y=-x+11 \\ y=11-x \end{gathered}`

Therefore, the equation for the perpendicular line passing through the point (8,3) is;

`y=11-x`