Chapter 4.3, Problem 48HP

To determine

When will the two lines be parallel or perpendicular, also find the equation of line that is parallel and perpendicular to line $y=\frac{3}{2}x-1$ .

Answer to Problem 48HP

The two lines will be parallel if there slopes are equal whereas, the lines will be perpendicular if slope of one line is negative reciprocal of another. And equation of parallel line is $y=\frac{3}{2}x+k$ whereas, equation of perpendicular line is $y=-\frac{2}{3}x+k$ for any $\left(k\right)$ belongs to real.

Explanation of Solution

Given information: The given line is $y=\frac{3}{2}x-1$ .

Calculation: The two lines will be parallel if there slopes are equal whereas, the lines will be perpendicular if slope of one line is negative reciprocal of another.

The line parallel to line $y=\frac{3}{2}x-1$ will have same slope $m=\frac{3}{2}$ . Therefore, any line of the form $y=\frac{3}{2}x+k$ having slope $m=\frac{3}{2}$ will be parallel to line $y=\frac{3}{2}x-1$ , where $\left(k\right)$ is any constant. So, consider a line $y=\frac{3}{2}x+k$ for $\left(k\in R\right)$ will be a line parallel to line $y=\frac{3}{2}x-1$ . Similarly, any line perpendicular to line $y=\frac{3}{2}x-1$ will have slope $\left(m\text{'}=-\frac{2}{3}\right)$ . Therefore, the line of the form $y=-\frac{2}{3}x+k$ will be perpendicular to line $y=\frac{3}{2}x-1$ for any $k\in R$ . Hence, the line of the form $y=-\frac{2}{3}x+k$ will be perpendicular to line $y=\frac{3}{2}x-1$ .