Question

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

$3x-2y=1$;$2x-3y=2$

Short answer

Neither parallel nor perpendicular.

Step-by-step solution

Step 1. Given information 

The given equations are$3x-2y=1$and$2x-3y=2$.

Step 2. Solve for y.

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

$$\begin{gathered} 3x-2y=1 \\ -2y=1-3x \\ \frac{-2y}{-2}=\frac{1-3x}{-2} \\ y=-\frac{1}{2}+\frac{3}{2}x \end{gathered}$$

$$\begin{gathered} 2x-3y=2 \\ -3y=2-2x \\ \frac{-3y}{-3}=\frac{2-2x}{-3} \\ y=-\frac{2}{3}+\frac{2}{3}x \end{gathered}$$

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 $y=mx+b$.

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

• The obtained slopes are neither same nor negative reciprocal of each other. So, the given lines are neither parallel nor perpendicular.