To identify which of the given points satisfy the system of equations, we need to solve the system:
[tex]\[
\begin{array}{l}
2x + y = 5 \\
3y = 15 - 6x
\end{array}
\][/tex]
First, solve the second equation for [tex]\( y \)[/tex]:
[tex]\[
3y = 15 - 6x
\][/tex]
Divide both sides by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 5 - 2x
\][/tex]
Next, substitute [tex]\( y = 5 - 2x \)[/tex] into the first equation:
[tex]\[
2x + (5 - 2x) = 5
\][/tex]
Combine like terms:
[tex]\[
2x + 5 - 2x = 5
\][/tex]
[tex]\[
5 = 5
\][/tex]
This is always true, which means the two equations are essentially the same and the system has infinitely many solutions along the line [tex]\( y = 5 - 2x \)[/tex].
We now need to check which of the given points lie on this line:
1. For the point [tex]\( (6, -7) \)[/tex]:
[tex]\[
y = 5 - 2x \implies -7 = 5 - 2(6) \implies -7 = 5 - 12 \implies -7 = -7
\][/tex]
This point satisfies the equation.
2. For the point [tex]\( (2, 1) \)[/tex]:
[tex]\[
y = 5 - 2x \implies 1 = 5 - 2(2) \implies 1 = 5 - 4 \implies 1 = 1
\][/tex]
This point satisfies the equation.
3. For the point [tex]\( (-2, -9) \)[/tex]:
[tex]\[
y = 5 - 2x \implies -9 = 5 - 2(-2) \implies -9 = 5 + 4 \implies -9 = 9
\][/tex]
This point does not satisfy the equation.
4. For the point [tex]\( (-4, 13) \)[/tex]:
[tex]\[
y = 5 - 2x \implies 13 = 5 - 2(-4) \implies 13 = 5 + 8 \implies 13 = 13
\][/tex]
This point satisfies the equation.
After checking all the points, the solutions to the system of equations among the given points are:
[tex]\[
(6, -7), (2, 1), \text{and} (-4, 13)
\][/tex]
