We are solving a system of equations using graphing. The system of equations is:

1. $-2x - 3y = -12$
2. $x + 3y = 9$

Step 1: Rewrite both equations in slope-intercept form $y = mx + b$.

Equation 1: $-2x - 3y = -12$

Solve for $y$:
$-3y = 2x - 12 \implies y = -\frac{2}{3}x + 4$

Equation 2: $x + 3y = 9$

Solve for $y$:
$3y = -x + 9 \implies y = -\frac{1}{3}x + 3$

The equations in slope-intercept form are:

1. $y = -\frac{2}{3}x + 4$
2. $y = -\frac{1}{3}x + 3$

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Step 2: Graph both equations.

• The first equation, $y = -\frac{2}{3}x + 4$, has a y-intercept of $4$ and a slope of $-\frac{2}{3}$. This means for every $3$ units to the right, the line goes down $2$ units.
• The second equation, $y = -\frac{1}{3}x + 3$, has a y-intercept of $3$ and a slope of $-\frac{1}{3}$. This means for every $3$ units to the right, the line goes down $1$ unit.

By graphing these, we determine if the lines intersect, are parallel, or are the same line. Let’s calculate the intersection point algebraically to verify.

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Step 3: Solve for the intersection point.

To find the intersection point of the lines, set the equations equal to each other:
$-\frac{2}{3}x + 4 = -\frac{1}{3}x + 3$
Simplify:
$-\frac{2}{3}x + \frac{1}{3}x = 3 - 4 \implies -\frac{1}{3}x = -1$
Multiply through by $-3$:
$x = 3$

Substitute $x = 3$ into either equation to find $y$:
Using $y = -\frac{1}{3}x + 3$:
$y = -\frac{1}{3}(3) + 3 = -1 + 3 = 2$

The solution is $(3, 2)$.

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Step 4: Interpret the result.

• Since the lines intersect at $(3, 2)$, there is exactly one solution.
• Correct choice: A. There is exactly one solution. The solution set is $\{(3, 2)\}$.

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Final Answer:

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

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Follow-up Questions:

1. What would happen if the lines were parallel?
2. How can you determine the slopes directly from the original equations?
3. What is the significance of the intersection point?
4. Why is slope-intercept form helpful for graphing?
5. How could you check the solution graphically?

Tip: When solving by graphing, always verify by substituting the intersection point into both equations to check correctness.