To solve the system of equations:

\[
\begin{aligned}
1. & \quad x + y = 5 \quad \text{(Equation 1)} \\
2. & \quad -x + 2y = 4 \quad \text{(Equation 2)}
\end{aligned}
\]

We can use either the substitution method or the elimination method. Here, I'll use the elimination method.

### Step 1: Add the two equations

First, we add Equation 1 and Equation 2 to eliminate \( x \):

\[
(x + y) + (-x + 2y) = 5 + 4
\]

This simplifies to:

\[
y + 2y = 9
\]

\[
3y = 9
\]

### Step 2: Solve for \( y \)

Now, solve for \( y \):

\[
y = \frac{9}{3} = 3
\]

### Step 3: Substitute \( y = 3 \) back into one of the original equations

Use Equation 1 to find \( x \):

\[
x + y = 5
\]

\[
x + 3 = 5
\]

\[
x = 5 - 3 = 2
\]

### Final Solution
The solution to the system of equations is:

\[
x = 2, \quad y = 3
\]

Would you like to go over any details or have any further questions?

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Here are some related questions you might find interesting:
1. What happens if we multiply Equation 1 by a constant before using the elimination method?
2. How would you solve this system using the substitution method instead?
3. Can you graphically represent the solution to these equations?
4. What would the solution be if one of the equations were non-linear?
5. How would this problem change if you had three variables instead of two?

**Tip:** When solving systems of equations, it's useful to double-check your solution by substituting the values back into the original equations.

Learn how to solve the system of equations x + y = 5 and -x + 2y = 4 using the elimination method. This problem covers fundamental algebraic concepts, including systems of linear equations and methods for solving them. Suitable for students in Grades 8-10.

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