Question

solve the system of equations by graphing.
\

$$\begin{cases} y = -3x \\\\ 3x - 2y = -18 \\end{cases}$$

Explanation:

Step1: Analyze the first equation

The first equation is \( y = -3x \), which is a linear equation in slope - intercept form (\( y=mx + b \)) where the slope \( m=-3 \) and the y - intercept \( b = 0 \). To graph this line, we know it passes through the origin \((0,0)\). We can find another point by plugging in \( x = 1 \), then \( y=-3(1)=-3 \), so the point \((1, - 3)\) is also on this line.

Step2: Rewrite the second equation in slope - intercept form

Start with \( 3x-2y=-18 \).
Subtract \( 3x \) from both sides: \( - 2y=-3x - 18 \).
Divide both sides by \(-2\): \( y=\frac{3}{2}x + 9 \).
This is a linear equation with slope \( m = \frac{3}{2}\) and y - intercept \( b = 9 \). To graph this line, we start at the y - intercept \((0,9)\). Then, using the slope (rise over run), from \((0,9)\), we rise 3 units and run 2 units to the right to get the point \((2,12)\) (or we can run 2 units to the left and fall 3 units to get the point \((- 2,6)\)).

Step3: Find the intersection point

The solution to the system of equations is the point where the two lines intersect. We can also solve the system algebraically to find the intersection point (since graphing might have some inaccuracies).
Substitute \( y=-3x \) into the second equation \( 3x-2y=-18 \):
\( 3x-2(-3x)=-18 \)
Simplify the left - hand side: \( 3x + 6x=-18 \)
Combine like terms: \( 9x=-18 \)
Divide both sides by 9: \( x=-2 \)
Now substitute \( x = - 2 \) into \( y=-3x \), we get \( y=-3(-2)=6 \)

Answer:

The solution to the system of equations is \( x=-2,y = 6 \) or the ordered pair \((-2,6)\)