Let's solve the given system of equations using the elimination method step-by-step.
We are given the system of equations:
[tex]\[
\begin{array}{l}
3x + 2y = 7 \quad \text{(Equation 1)} \\
-9x - 6y = -21 \quad \text{(Equation 2)}
\end{array}
\][/tex]
### Step 1: Align the Coefficients
To eliminate one of the variables, we first align the coefficients of either [tex]\(x\)[/tex] or [tex]\(y\)[/tex]. Here, we notice that multiplying Equation 1 by 3 will align the coefficients of [tex]\(x\)[/tex]:
[tex]\[
3 \times (3x + 2y) = 3 \times 7
\][/tex]
[tex]\[
9x + 6y = 21 \quad \text{(New Equation 1)}
\][/tex]
Equation 2 remains the same:
[tex]\[
-9x - 6y = -21 \quad \text{(Equation 2)}
\][/tex]
### Step 2: Add the Equations
Next, we add the New Equation 1 and Equation 2 to eliminate [tex]\(x\)[/tex]:
[tex]\[
(9x + 6y) + (-9x - 6y) = 21 + (-21)
\][/tex]
This simplifies to:
[tex]\[
0x + 0y = 0
\][/tex]
[tex]\[
0 = 0
\][/tex]
### Step 3: Analyze the Result
The result of [tex]\(0 = 0\)[/tex] is always true. This indicates that the two original equations are actually the same line when simplified proportionally, meaning they overlap entirely.
### Conclusion
Since the equations represent the same line, there are infinitely many solutions. In other words, every point on the line defined by [tex]\(3x + 2y = 7\)[/tex] will satisfy both equations.
Thus, the solution to the system of equations is:
```
There are infinitely many solutions.
```
