https://file.aiquickdraw.com/mathbot/file/1727203083656ltwf9upr.jpg

Let's analyze the system of equations and the graph provided to solve for \( x \) and \( y \).

The system of equations given is:
\[ 
\begin{cases}
y = -2x + 7 \\
y = 5x - 7 
\end{cases}
\]

To find the solution to this system of equations, we can set the two equations equal to each other since they both equal \( y \):

\[ -2x + 7 = 5x - 7 \]

Now, solve for \( x \):

1. Combine like terms:
\[ 7 + 7 = 5x + 2x \]
\[ 14 = 7x \]

2. Divide both sides by 7:
\[ x = 2 \]

Now substitute \( x = 2 \) back into one of the original equations to find \( y \). Using the first equation:

\[ y = -2(2) + 7 \]
\[ y = -4 + 7 \]
\[ y = 3 \]

Thus, the solution to the system of equations is:
\[ x = 2 \]
\[ y = 3 \]

These values are the coordinates of the intersection point of the two lines. According to the graph, the intersection point at \((2, 3)\) confirms our solution.

Would you like more details or have any questions about the solution? Here are some related questions to consider:

1. How do you solve a system of equations using the substitution method?
2. What is the graphical interpretation of solving a system of equations?
3. How can you verify the solution of a system of equations algebraically?
4. What are the different methods to solve a system of linear equations?
5. How can you interpret the solution to a system of equations in a real-world context?

**Tip:** When solving systems of equations graphically, always check the intersection point to ensure it satisfies both equations.

Solve the system of equations y = -2x + 7 and y = 5x - 7. Determine the values of x and y.

Learn how to solve the system of linear equations y = -2x + 7 and y = 5x - 7. This problem covers key algebraic concepts such as linear equations, solving systems of equations, and graphing techniques. Suitable for students in Grades 8-10.

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