Question

solve the system of equations by graphing.\

$$\begin{cases} y = -3x - 6 \\\\ 3x - 2y = -24 \\end{cases}$$

Explanation:

Step1: Analyze the first equation

The first equation is \( y = -3x - 6 \). This is in slope - intercept form (\( y=mx + b \)), where the slope \( m=-3 \) and the y - intercept \( b = - 6 \). To graph this line, we can start by plotting the y - intercept at the point \( (0,-6) \). Then, using the slope (rise over run), since the slope is - 3 (or \( \frac{-3}{1} \)), we can go down 3 units and to the right 1 unit from the y - intercept to get another point, for example, \( (1,-9) \), or up 3 units and to the left 1 unit to get \( (-1,-3) \).

Step2: Rewrite the second equation in slope - intercept form

We have the second equation \( 3x-2y=-24 \). We want to solve for \( y \):

$$
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$$

This is in slope - intercept form with slope \( m = \frac{3}{2} \) and y - intercept \( b = 12 \). To graph this line, we plot the y - intercept at \( (0,12) \). Then, using the slope \( \frac{3}{2} \) (rise 3, run 2), we can go up 3 units and to the right 2 units from the y - intercept to get a new point, such as \( (2,15) \), or down 3 units and to the left 2 units to get \( (-2,9) \).

Step3: Find the intersection point of the two lines

When we graph both lines, we look for the point where they intersect. We can also solve the system algebraically to find the intersection point (which should match the graph's intersection). Let's substitute \( y=-3x - 6 \) into \( 3x-2y=-24 \):

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$$

Now substitute \( x = - 4 \) into \( y=-3x - 6 \):
\( y=-3(-4)-6=12 - 6 = 6 \)

So the point of intersection of the two lines (the solution to the system) is \( (-4,6) \). When we graph the two lines, the first line \( y=-3x - 6 \) is a line with a negative slope, and the second line \( y=\frac{3}{2}x + 12 \) is a line with a positive slope. The two lines will cross at the point \( (-4,6) \).

Answer:

The solution to the system of equations is \( x=-4,y = 6 \) or the ordered pair \( (-4,6) \)