Question

solve the system by graphing.\

$$\begin{cases}-2x - 3y = -12\\\\x + 3y = 9\\end{cases}$$

\dots\
use the graphing tool to graph the system.\
click to enlarge graph\
select the correct choice below and, if necessary, fill in the answer box to complete your choice.\
\bigcirc a. there is exactly one solution. the solution set is \\{\\ \\}.\
(simplify your answer. type an ordered pair.)\
\bigcirc b. there are infinitely many solutions. the solution set is \\{\\(x,y\\)| - 2x - 3y = -12\\} or \\{\\(x,y\\)|x + 3y = 9\\}.\
\bigcirc c. the solution set is \varnothing.

Explanation:

Step1: Analyze the system of equations

We have the system:
\[

$$\begin{cases} -2x - 3y = -12 \\ x + 3y = 9 \end{cases}$$

\]
We can solve this system by adding the two equations together to eliminate \(y\).

Step2: Add the two equations

Adding the left - hand sides and the right - hand sides of the two equations:
\((-2x - 3y)+(x + 3y)=-12 + 9\)
Simplify the left - hand side: \(-2x-3y + x + 3y=-x\)
Simplify the right - hand side: \(-12 + 9=-3\)
So we get \(-x=-3\), which implies \(x = 3\).

Step3: Substitute \(x = 3\) into one of the equations

Substitute \(x = 3\) into the second equation \(x + 3y=9\):
\(3+3y = 9\)
Subtract 3 from both sides: \(3y=9 - 3=6\)
Divide both sides by 3: \(y = 2\)

Answer:

A. There is exactly one solution. The solution set is \((3,2)\)