Question

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

$y=\frac{3}{4}x-3$;$3x-4y=-2$

Short answer

The given lines are parallel.

Step-by-step solution

Step 1. Given information

The given equations are $y=\frac{3}{4}x-3$and$3x-4y=-2$.

Step 2. Solve for y.

The first given equation is already in the slope-intercept form, so solve the second equation for $y$.

$\begin{array}{rcl}3x-4y& =& -2\\ -4y& =& -2-3x\\ \frac{-4y}{-4}& =& \frac{-2-3x}{-4}\\ y& =& \frac{1}{2}+\frac{3}{4}x\end{array}$

Step 3. Determine slope and y- intercepts.

• Now both equations are in slope-intercept form.
• Determine the slope by comparing both equations with $y=mx+b$.

$m=\frac{3}{4}$

• The y-intercept of first and second line is$\left(0,-3\right)$and$\left(0,\frac{1}{2}\right)$respectively.

Step 4. Conclusion

Since both the lines have same slope but distinct values of y-intercepts, they are parallel.