Q3. - Algebra 1

Question

Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

Which equation of a line is perpendicular to $y=\frac{3}{5}x-3$.

1. $y=-\frac{5}{3}x+2$
2. $y=\frac{5}{3}x-2$
3. $y=-\frac{3}{5}x+2$
4. $y=\frac{3}{5}x-2$

Step-by-Step Solution

Answer

The equation of a line which is perpendicular to the line $y=\frac{3}{5}x-3$ is $y=-\frac{5}{3}x+2$. Therefore, the option A is correct.

Step 1. Write the slope-intercept form of the line.

The slope-intercept form of the line is:

$y=mx+c$, where $m$ is the slope of the line and $c$ is the $y$-intercept of the line.

Step 2. Determine the equation of a line which is perpendicular to the line y = 3 5 x − 3 .

Compare the equation of the line $y=\frac{3}{5}x-3$ with the equation $y=mx+c$.

Therefore, it is obtained that:

$m=\frac{3}{5}$.

Therefore, the slope of the line $y=\frac{3}{5}x-3$ is $\frac{3}{5}$.

Compare the equation of the line $y=\frac{-5}{3}x+2$ with the equation $y=mx+c$.

Therefore, it is obtained that:

$m=\frac{-5}{3}$.

Therefore, the slope of the line $y=\frac{-5}{3}x+2$ is $\frac{-5}{3}$.

Compare the equation of the line  $y=\frac{5}{3}x-2$ with the equation $y=mx+c$.

Therefore, it is obtained that:

$m=\frac{5}{3}$.

Therefore, the slope of the line $y=\frac{5}{3}x-2$ is $\frac{5}{3}$.

Compare the equation of the line $y=-\frac{3}{5}x+2$ with the equation $y=mx+c$.

Therefore, it is obtained that:

$m=-\frac{3}{5}$.

Therefore, the slope of the line $y=-\frac{3}{5}x+2$ is $-\frac{3}{5}$.

Compare the equation of the line $y=\frac{3}{5}x-2$ with the equation $y=mx+c$.

Therefore, it is obtained that:

$m=\frac{3}{5}$.

Therefore, the slope of the line $y=\frac{3}{5}x-2$ is $\frac{3}{5}$.

 

Now it can be noticed that:

$\begin{array}{c}\text{slope of line }\left(y=\frac{3}{5}x-3\right)\times \text{slope of line }\left(y=\frac{-5}{3}x+2\right)=\frac{3}{5}\times \frac{-5}{3}\\ =-1\end{array}$

$\begin{array}{c}\text{slope of line }\left(y=\frac{3}{5}x-3\right)\times \text{slope of line }\left(y=\frac{5}{3}x-2\right)=\frac{3}{5}\times \frac{5}{3}\\ =1\end{array}$

$\begin{array}{c}\text{slope of line }\left(y=\frac{3}{5}x-3\right)\times \text{slope of line }\left(y=-\frac{3}{5}x+2\right)=\frac{3}{5}\times \frac{-3}{5}\\ =\frac{-9}{25}\end{array}$

$\begin{array}{c}\text{slope of line }\left(y=\frac{3}{5}x-3\right)\times \text{slope of line }\left(y=\frac{3}{5}x-2\right)=\frac{3}{5}\times \frac{3}{5}\\ =\frac{9}{25}\end{array}$

It is known that if the two lines are perpendicular then the product of the slopes of the lines is $-1$.

It can be noticed as the product of the slopes of the lines $y=\frac{3}{5}x-3$ and $y=\frac{-5}{3}x+2$ is $-1$. Therefore the lines $y=\frac{3}{5}x-3$ and $y=\frac{-5}{3}x+2$ are perpendicular.

Therefore, the equation of a line which is perpendicular to the line $y=\frac{3}{5}x-3$ is $y=\frac{-5}{3}x+2$. Therefore, the option A is correct.