Sure! Let's solve the given system of equations using the elimination method. The system of equations is:
[tex]\[
\begin{array}{l}
3x + 2y = 7 \\
-9x - 6y = -21 \\
\end{array}
\][/tex]
Step 1: Write the system of equations.
[tex]\[
\begin{array}{l}
3x + 2y = 7 \\
-9x - 6y = -21 \\
\end{array}
\][/tex]
Step 2: Identify a way to eliminate one of the variables.
Notice that the second equation is a multiple of the first equation. To see this more clearly, let's rewrite the second equation:
[tex]\[
-9x - 6y = -21
\][/tex]
We can observe that \(-9\) is \(-3\) times \(3\), \(-6\) is \(-3\) times \(2\), and \(-21\) is \(-3\) times \(7\).
Step 3: Check for consistency between the equations.
The second equation is exactly \(-3\) times the first equation:
[tex]\[
-3(3x + 2y) = -3 \cdot 7\\
\Rightarrow -9x - 6y = -21
\][/tex]
This confirms that the second equation is a multiple of the first.
Step 4: Interpret the results.
Since both equations describe the same relationship (one is just a scalar multiple of the other), these two lines overlap completely. Therefore, the system of equations is dependent and has infinitely many solutions. This means that any solution \((x, y)\) that satisfies the first equation will also satisfy the second equation. Consequently, there are infinitely many solutions to this system of equations.
Thus, the system is dependent and consistent, having infinite solutions.
