Question

Consider the following equation determine if the two lines are perpendicular

Answer 1

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

When the variable y is isolated and the expressions are reduced to its lowest terms, the coefficient of the variable x corresponds to the slope of the line.

Isolate y from both equations to find if the lines are perpendicular or not.

First expression:
`\begin{gathered} 3-(5y-x)/(2)=2x+2 \\ \\ \Rightarrow-(5y-x)/(2)=2x-1 \\ \\ \Rightarrow5y-x=-2(2x-1) \\ \\ \Rightarrow5y-x=-4x+2 \\ \\ \Rightarrow5y=-3x+2 \\ \\ \Rightarrow y=-(3)/(5)x+(2)/(5) \end{gathered}`Second expression:
`\begin{gathered} 3x-5y=13 \\ \\ \Rightarrow5y=3x-13 \\ \\ \Rightarrow y=(3)/(5)x-(13)/(5) \end{gathered}`

The slope of the first line is -3/5 and the slope of the second line is 3/5. The product of the slopes is:

`\left(-(3)/(5)\right)\left((3)/(5)\right)=-(9)/(25)`

Which is not equal to -1.

Therefore, the lines are not perpendicular.