Question

In the following exercise, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

$2x+3y=5$;$3x-2y=7$

Short answer

The given lines are perpendicular.

Step-by-step solution

Step 1. Given information

The given equations are$2x+3y=5$and$3x-2y=7$.

Step 2. Solve for y.

Solve both the equations to obtain the value of y in terms of x.

$\begin{array}{rcl}2x+3y& =& 5\\ 3y& =& 5-2x\\ \frac{3y}{3}& =& \frac{5-2x}{3}\\ y& =& \frac{5}{3}-\frac{2}{3}x\\ & & \end{array}$

$\begin{array}{rcl}3x-2y& =& 7\\ -2y& =& 7-3x\\ \frac{-2y}{-2}& =& \frac{7-3x}{-2}\\ y& =& -\frac{7}{2}+\frac{3}{2}x\end{array}$

Step 3. Determine slopes.

• Now both equations are in slope-intercept form.
• Determine the slopes $m_1$and $m_2$by comparing both equations with data-custom-editor="chemistry" $\mathrm{y}=\mathrm{mx}+\mathrm{b}$.

$$\begin{gathered} m_1=-\frac{2}{3} \\ {m}_2=\frac{3}{2} \end{gathered}$$

• The obtained slopes are negative reciprocal of each other. So, the given lines are perpendicular lines.

Step 4. Check

• Multiply both slopes to check the answer.

$\begin{array}{rcl}{m}_1{m}_2& =& \left(-\frac{2}{3}\right)\left(\frac{3}{2}\right)\\ & =& -1\end{array}$

• The product of slopes is -1. So, the obtained result is correct.