Chapter 3, Problem 5CR

Find the equation of the line perpendicular to the line $y=2x+1$ and containing the point $\left(3,5\right)$. Express your answer in slope-intercept form and graph the line.

To determine

To calculate: The equation of the line perpendicular to the line $y=2x+1$ and containing the point $\left(3,5\right)$.

Answer to Problem 5CR

Solution:

The equation of the line perpendicular to the line $y=2x+1$ and containing the point $\left(3,5\right)$ is $Y=-\frac{1}{2}x+\frac{13}{2}$.

Explanation of Solution

Given:

The given line is $y=2x+1$.

Formula used:

The product of the slopes of 2 perpendicular lines is always $-1$.

Calculation:

Here, we have the line $y=2x+1$, with slope $m=2$.

Thus, for the line perpendicular to the given line, the slope will be $m_1=-\frac{1}{2}$.

Thus, the equation of the line becomes

$Y=-\frac{1}{2}x+c$

Now, we need to find the value of $c$.

Thus, we can now plug in the points $(x,y)=(3,5)$ in the above equation.

Therefore, we get

$5=-\frac{1}{2}(3)+c$

$\Rightarrow c=5+\frac{3}{2}=\frac{13}{2}$

Therefore, the equation of the line perpendicular to the given line and passing through the given point is

$Y=-\frac{1}{2}x+\frac{13}{2}$.