Question

1. maria solved the system of equations \\( \

$$\begin{cases} x - y = 6 \\\\ y = -2x + 3 \\end{cases}$$

\\) by graphing. her graph is shown below. she determined that the solution is \\( (3, -3) \\).
a coordinate plane with the following attributes:

• two intersecting lines.
• one line passes through the points \\( (0, 3) \\), \\( (3, -3) \\), and \\( (6, -9) \\).
• the other line passes through the points \\( (0, -6) \\), \\( (3, -3) \\), and \\( (6, 0) \\).

Explanation:

Step1: Analyze the first equation

The first equation (assuming it's \(x - y = 6\), since the original has a typo but we can infer from the points) can be rewritten as \(y=x - 6\). Let's check if \((3,-3)\) satisfies this. Substitute \(x = 3\), \(y=-3\): \( - 3=3 - 6\)? \(3 - 6=-3\), so it does for this rewritten form (or if the first equation was \(x - y=6\)).

Step2: Analyze the second equation

The second equation is \(y=-2x + 3\). Substitute \(x = 3\), \(y=-3\): \( - 3=-2(3)+3\)? \( - 2(3)+3=-6 + 3=-3\), which matches. But wait, the lines: one line passes through \((0,3),(3,-3),(6,-9)\) – let's check its slope. Slope \(m=\frac{-3 - 3}{3 - 0}=\frac{-6}{3}=-2\), so equation \(y=-2x + 3\) (since when \(x = 0\), \(y = 3\)), correct. The other line passes through \((0,-6),(3,-3),(6,0)\): slope \(m=\frac{-3+6}{3 - 0}=\frac{3}{3}=1\), equation \(y=x - 6\) (when \(x = 0\), \(y=-6\)). Now, find intersection: solve \(y=x - 6\) and \(y=-2x + 3\). Set equal: \(x - 6=-2x + 3\), \(3x=9\), \(x = 3\), then \(y=3 - 6=-3\). Wait, but Maria's solution is \((3,-3)\), but let's check the lines' intersection. The two lines: one with slope -2 (equation \(y=-2x + 3\)) and one with slope 1 (equation \(y=x - 6\)) intersect at \(x = 3\), \(y=-3\)? Wait, no: \(y=x - 6\) at \(x = 3\) is \(y=-3\), \(y=-2x + 3\) at \(x = 3\) is \(y=-3\). But wait, the line \(y=x - 6\) passes through \((0,-6),(3,-3),(6,0)\), and \(y=-2x + 3\) passes through \((0,3),(3,-3),(6,-9)\). So their intersection is \((3,-3)\)? Wait, but let's check if Maria made a mistake? Wait no, the calculation shows that \((3,-3)\) is the solution. Wait, maybe the original first equation was \(x - y=6\) (since \(x - y=6\) gives \(y=x - 6\), which matches the line through \((0,-6),(3,-3),(6,0)\)). So the solution \((3,-3)\) is correct? Wait, no, wait: \(x - y=6\) and \(y=-2x + 3\). Solving: substitute \(y\) into first equation: \(x-(-2x + 3)=6\), \(x + 2x-3=6\), \(3x=9\), \(x = 3\), \(y=-2(3)+3=-3\). So the solution is correct. Wait, but maybe the question is about whether Maria's solution is correct? Wait, the problem (probably) is to check if her solution is correct. So steps: 1. Find equations of both lines. 2. Solve the system algebraically. 3. Compare with her solution.

Answer:

The solution \((3, - 3)\) is correct because substituting \(x = 3\) and \(y=-3\) into both equations satisfies them, and solving the system algebraically (by equating the two linear equations) also gives \(x = 3\) and \(y=-3\).