https://file.aiquickdraw.com/mathbot/file/1728243445566pspihl92.jpg

To determine which system has \((3, 3)\) as a solution, we need to substitute \(x = 3\) and \(y = 3\) into each system of equations and check which one holds true.

### First system:
\[
\begin{aligned}
-2x - 2y &= 2 \\
-4x - 3y &= 2
\end{aligned}
\]
Substitute \(x = 3\) and \(y = 3\):
1. \( -2(3) - 2(3) = -6 - 6 = -12 \neq 2 \)
Since the first equation does not hold, this system does not have \((3, 3)\) as a solution.

---

### Second system:
\[
\begin{aligned}
-x + y &= 0 \\
x + 6y &= 21
\end{aligned}
\]
Substitute \(x = 3\) and \(y = 3\):
1. \( -(3) + 3 = 0 \quad \text{(True)} \)
2. \( 3 + 6(3) = 3 + 18 = 21 \quad \text{(True)} \)

Both equations are true, so this system does have \((3, 3)\) as a solution.

---

### Third system:
\[
\begin{aligned}
-6x + 3y &= 6 \\
-x - 5y &= 23
\end{aligned}
\]
Substitute \(x = 3\) and \(y = 3\):
1. \( -6(3) + 3(3) = -18 + 9 = -9 \neq 6 \)
Since the first equation does not hold, this system does not have \((3, 3)\) as a solution.

---

### Conclusion:
The second system:
\[
\begin{aligned}
-x + y &= 0 \\
x + 6y &= 21
\end{aligned}
\]
has the solution \((3, 3)\).

Would you like more details or have any questions?

Here are five related questions you can explore:
1. How can we graphically interpret the solution of a system of equations?
2. How do we check if a system of equations is consistent or inconsistent?
3. Can a system have more than one solution?
4. How would you use substitution to solve a system like this?
5. What is the difference between dependent and independent systems?

**Tip:** Always substitute the values back into all equations to verify that the solution satisfies each equation in the system.

Determine which system of linear equations has the solution (3, 3) by substituting the values into each equation. This step-by-step algebraic solution explains how to verify consistency in linear systems. Suitable for students in Grades 8-10.

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