To determine which point is a solution to the system of linear equations, we need to check each point against the given equations and see which one satisfies both equations:
[tex]\[
\begin{array}{l}
y = -x + 2 \\
3x - y = 6
\end{array}
\][/tex]
The points given are [tex]\((-1, 3)\)[/tex], [tex]\((0, 2)\)[/tex], [tex]\((2, 0)\)[/tex], and [tex]\((3, -1)\)[/tex]. We will test each point step-by-step.
1. Point [tex]\((-1, 3)\)[/tex]:
- For [tex]\(y = -x + 2\)[/tex]:
[tex]\[
3 = -(-1) + 2 \implies 3 = 1 + 2 \implies 3 = 3 \quad \text{(True)}
\][/tex]
- For [tex]\(3x - y = 6\)[/tex]:
[tex]\[
3(-1) - 3 = 6 \implies -3 - 3 = 6 \implies -6 = 6 \quad \text{(False)}
\][/tex]
This point does not satisfy the second equation.
2. Point [tex]\((0, 2)\)[/tex]:
- For [tex]\(y = -x + 2\)[/tex]:
[tex]\[
2 = -0 + 2 \implies 2 = 2 \quad \text{(True)}
\][/tex]
- For [tex]\(3x - y = 6\)[/tex]:
[tex]\[
3(0) - 2 = 6 \implies 0 - 2 = 6 \implies -2 = 6 \quad \text{(False)}
\][/tex]
This point does not satisfy the second equation either.
3. Point [tex]\((2, 0)\)[/tex]:
- For [tex]\(y = -x + 2\)[/tex]:
[tex]\[
0 = -2 + 2 \implies 0 = 0 \quad \text{(True)}
\][/tex]
- For [tex]\(3x - y = 6\)[/tex]:
[tex]\[
3(2) - 0 = 6 \implies 6 - 0 = 6 \implies 6 = 6 \quad \text{(True)}
\][/tex]
This point satisfies both equations.
4. Point [tex]\((3, -1)\)[/tex]:
- For [tex]\(y = -x + 2\)[/tex]:
[tex]\[
-1 = -3 + 2 \implies -1 = -1 \quad \text{(True)}
\][/tex]
- For [tex]\(3x - y = 6\)[/tex]:
[tex]\[
3(3) - (-1) = 6 \implies 9 + 1 = 6 \implies 10 = 6 \quad \text{(False)}
\][/tex]
This point does not satisfy the second equation.
After checking all the points, we can conclude that the point [tex]\((2, 0)\)[/tex] is the only solution that satisfies both equations in the system.
Thus, the solution to the system of linear equations is:
[tex]\[
(2, 0)
\][/tex]
