Question

which graph represents this system?
3x + 2y = -6
$y = -\frac{3}{2}x + 2$

Explanation:

Step1: Rewrite the first equation in slope - intercept form

We start with the equation \(3x + 2y=-6\). To get it in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept), we solve for \(y\).
Subtract \(3x\) from both sides: \(2y=-3x - 6\).
Then divide each term by 2: \(y=-\frac{3}{2}x-3\).

Step2: Analyze the slopes and y - intercepts of the two lines

The first line (from \(3x + 2y=-6\)) has the equation \(y =-\frac{3}{2}x-3\), so its slope \(m_1=-\frac{3}{2}\) and its y - intercept \(b_1=-3\) (the line crosses the y - axis at \((0, - 3)\)).
The second line has the equation \(y=-\frac{3}{2}x + 2\), so its slope \(m_2=-\frac{3}{2}\) and its y - intercept \(b_2 = 2\) (the line crosses the y - axis at \((0,2)\)).

Since the slopes of the two lines (\(m_1=m_2 =-\frac{3}{2}\)) are equal and the y - intercepts (\(b_1=-3\) and \(b_2 = 2\)) are different, the two lines are parallel.

To identify the graph:

• One line should have a y - intercept at \((0, - 3)\) and the other at \((0,2)\).
• Both lines should have a slope of \(-\frac{3}{2}\) (they should be decreasing from left to right, and for every 2 units we move to the right, we move down 3 units).

Answer:

The graph that represents the system will show two parallel lines. One line has a y - intercept at \((0,-3)\) (from \(y =-\frac{3}{2}x-3\)) and the other has a y - intercept at \((0,2)\) (from \(y=-\frac{3}{2}x + 2\)), and both lines have a slope of \(-\frac{3}{2}\).