To find which of the given points are solutions to the system of equations:
[tex]\[
\begin{array}{l}
2x + y = 5 \\
3y = 15 - 6x
\end{array}
\][/tex]
we will substitute each of the points [tex]\((x, y)\)[/tex] into both equations and see if both equations are satisfied.
### Checking the point [tex]\((6, -7)\)[/tex]:
1. Substitute [tex]\(x = 6\)[/tex] and [tex]\(y = -7\)[/tex] into the first equation [tex]\(2x + y = 5\)[/tex]:
[tex]\[
2(6) + (-7) = 12 - 7 = 5 \quad \text{(True)}
\][/tex]
2. Substitute [tex]\(x = 6\)[/tex] and [tex]\(y = -7\)[/tex] into the second equation [tex]\(3y = 15 - 6x\)[/tex]:
[tex]\[
3(-7) = 15 - 6(6) \implies -21 = 15 - 36 \implies -21 = -21 \quad \text{(True)}
\][/tex]
Since both equations are satisfied, [tex]\((6, -7)\)[/tex] is a solution.
### Checking the point [tex]\((2, 1)\)[/tex]:
1. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into the first equation [tex]\(2x + y = 5\)[/tex]:
[tex]\[
2(2) + 1 = 4 + 1 = 5 \quad \text{(True)}
\][/tex]
2. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 1\)[/tex] into the second equation [tex]\(3y = 15 - 6x\)[/tex]:
[tex]\[
3(1) = 15 - 6(2) \implies 3 = 15 - 12 \implies 3 = 3 \quad \text{(True)}
\][/tex]
Since both equations are satisfied, [tex]\((2, 1)\)[/tex] is a solution.
### Checking the point [tex]\((-2, -9)\)[/tex]:
1. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -9\)[/tex] into the first equation [tex]\(2x + y = 5\)[/tex]:
[tex]\[
2(-2) + (-9) = -4 - 9 = -13 \quad \text{(False)}
\][/tex]
Since the first equation is not satisfied, we don't need to check the second equation. [tex]\((-2, -9)\)[/tex] is not a solution.
### Checking the point [tex]\((-4, 13)\)[/tex]:
1. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 13\)[/tex] into the first equation [tex]\(2x + y = 5\)[/tex]:
[tex]\[
2(-4) + 13 = -8 + 13 = 5 \quad \text{(True)}
\][/tex]
2. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = 13\)[/tex] into the second equation [tex]\(3y = 15 - 6x\)[/tex]:
[tex]\[
3(13) = 15 - 6(-4) \implies 39 = 15 + 24 \implies 39 = 39 \quad \text{(True)}
\][/tex]
Since both equations are satisfied, [tex]\((-4, 13)\)[/tex] is a solution.
### Conclusion:
The points that are solutions to the given system of equations are:
[tex]\[
(6, -7), (2, 1), \text{and} (-4, 13)
\][/tex]
