Question

GRAPHING LINEAR SYSTEMS Use the graphing method to solve the linear system and describe its solution(s). $$ \begin{aligned} &3 x-2 y=0\\\ &3 x-2 y=-4 \end{aligned} $$

Short answer

The system of equations has no solution because the lines represented by the equations are parallel.

Step-by-step solution

Rearranging the equations into slope-intercept form

To begin with, both equations need to be rearranged into slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept. For the first equation, dividing all terms by -2 gives \(y = 1.5x\). For the second, adding \(3x\) to both sides and dividing by -2 gives \(y = 1.5x + 2\).

Graphing the equations

Now both equations can be graphed in the same coordinate plane. The equation \(y = 1.5x\) is a direct variation with a slope of 1.5 or \(3/2\) and passes through the origin. The equation \(y = 1.5x + 2\) is a linear equation with the same slope and y-intercept at 2. The graphs would therefore be two lines with the same slopes but different y-intercepts.

Finding the solution

Finally, since the lines don’t cross (they are parallel), the system doesn’t have a solution.