Question

solve the system of equations by graphing.\

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

Explanation:

Step1: Rewrite the second equation

Rewrite \(-3x - 2y = 24\) in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept).
First, isolate \(y\):
\(-2y=3x + 24\)
Divide both sides by \(-2\): \(y=-\frac{3}{2}x-12\)
The first equation is \(y = 3x+6\), with slope \(m_1 = 3\) and y - intercept \(b_1=6\).
The second equation is \(y=-\frac{3}{2}x - 12\), with slope \(m_2=-\frac{3}{2}\) and y - intercept \(b_2=- 12\).

Step2: Analyze the slopes and intercepts

For a system of linear equations \(y = m_1x + b_1\) and \(y=m_2x + b_2\), if \(m_1
eq m_2\), the lines intersect at one point. Here, \(m_1 = 3\) and \(m_2=-\frac{3}{2}\), so the lines are not parallel (since slopes are different) and not coincident (since slopes are different and y - intercepts are different).
To find the intersection point, we can also solve the system algebraically (since graphing can be done by plotting the y - intercepts and using the slopes to find another point on each line and then finding the intersection). But we can also substitute \(y = 3x + 6\) into the second equation:
Substitute \(y\) in \(-3x-2y = 24\):
\(-3x-2(3x + 6)=24\)
\(-3x-6x-12 = 24\)
\(-9x=24 + 12\)
\(-9x=36\)
\(x=-4\)
Then substitute \(x = - 4\) into \(y = 3x+6\):
\(y=3\times(-4)+6=-12 + 6=-6\)

Answer:

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