The problem is asking for the solution to the system of equations represented by the graph and the given equations:

$y = -2x + 4 \quad \text{and} \quad y = x - 5$

We are looking for the point where the two lines intersect, which represents the solution to the system of equations.

Solution by Substitution

We can solve the system algebraically by setting the two equations equal to each other since both are in the form $y =$:

$-2x + 4 = x - 5$

Now, let's solve for $x$:

1. Add $2x$ to both sides: $4 = 3x - 5$

2. Add $5$ to both sides: $9 = 3x$

3. Divide by 3: $x = 3$

Now substitute $x = 3$ into either equation to find $y$. Using $y = x - 5$:

$y = 3 - 5 = -2$

Thus, the solution is $(3, -2)$.

Final Answer:

The correct solution is A. (3, -2), which matches the point of intersection in the graph.

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. Solve the system of equations: $y = 2x + 1$ and $y = -x + 3$.
2. How can you verify graphically the solution to a system of equations?
3. What does it mean if two lines have no point of intersection?
4. Solve for $x$ and $y$: $y = 3x + 7$ and $y = 5x - 1$.
5. Can a system of equations have infinitely many solutions?

Tip: When solving systems of equations graphically, always look for the point where the lines intersect, as that represents the solution.