To determine for which system of equations the point [tex]\((2, 2)\)[/tex] is a solution, we need to substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 2\)[/tex] into each system's equations and check whether both equations hold true simultaneously.
### System A
[tex]\[
\begin{cases}
-3x + 3y = 0 \\
x + 6y = 10
\end{cases}
\][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 2\)[/tex] into the equations:
1. [tex]\(-3(2) + 3(2) = -6 + 6 = 0\)[/tex] ⟹ This equation holds true.
2. [tex]\(2 + 6(2) = 2 + 12 = 14\)[/tex] ⟹ This equation does not hold true (10 ≠ 14).
Therefore, System A is not satisfied.
### System B
[tex]\[
\begin{cases}
-2x + 5y = -6 \\
4x - 2y = 4
\end{cases}
\][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 2\)[/tex] into the equations:
1. [tex]\(-2(2) + 5(2) = -4 + 10 = 6\)[/tex] ⟹ This equation does not hold true (-6 ≠ 6).
2. [tex]\(4(2) - 2(2) = 8 - 4 = 4\)[/tex] ⟹ This equation holds true.
Therefore, System B is not satisfied.
### System C
[tex]\[
\begin{cases}
5x - 2y = -6 \\
3x - 4y = 2
\end{cases}
\][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 2\)[/tex] into the equations:
1. [tex]\(5(2) - 2(2) = 10 - 4 = 6\)[/tex] ⟹ This equation does not hold true (-6 ≠ 6).
2. [tex]\(3(2) - 4(2) = 6 - 8 = -2\)[/tex] ⟹ This equation does not hold true (2 ≠ -2).
Therefore, System C is not satisfied.
### System D
[tex]\[
\begin{cases}
2x + 3y = 10 \\
4x + 5y = 18
\end{cases}
\][/tex]
Substituting [tex]\(x = 2\)[/tex] and [tex]\(y = 2\)[/tex] into the equations:
1. [tex]\(2(2) + 3(2) = 4 + 6 = 10\)[/tex] ⟹ This equation holds true.
2. [tex]\(4(2) + 5(2) = 8 + 10 = 18\)[/tex] ⟹ This equation holds true.
Therefore, System D is satisfied.
So, the point [tex]\((2, 2)\)[/tex] is a solution for System D.
