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The given system of equations is:

\[
\begin{aligned}
2x + y &= 8 \\
3x - y &= 7
\end{aligned}
\]

We are instructed to use the echelon method (Gaussian elimination) to solve this system.

### Step 1: Convert the system into an augmented matrix.

\[
\begin{bmatrix}
2 & 1 & | & 8 \\
3 & -1 & | & 7
\end{bmatrix}
\]

### Step 2: Perform row operations to create an upper triangular matrix (echelon form).

**Row 1 (R1):** \(\begin{bmatrix} 2 & 1 & | & 8 \end{bmatrix}\)

**Row 2 (R2):** \(\begin{bmatrix} 3 & -1 & | & 7 \end{bmatrix}\)

We will first eliminate the \(x\)-term in Row 2. To do this, we multiply Row 1 by \(\frac{3}{2}\) and subtract it from Row 2:

\[
R2 \to R2 - \frac{3}{2} \times R1
\]

Performing the operation:

\[
\begin{aligned}
R1 &= \begin{bmatrix} 2 & 1 & | & 8 \end{bmatrix} \\
R2 &= \begin{bmatrix} 3 & -1 & | & 7 \end{bmatrix} - \frac{3}{2} \times \begin{bmatrix} 2 & 1 & | & 8 \end{bmatrix} \\
R2 &= \begin{bmatrix} 3 & -1 & | & 7 \end{bmatrix} - \begin{bmatrix} 3 & 1.5 & | & 12 \end{bmatrix} \\
R2 &= \begin{bmatrix} 0 & -2.5 & | & -5 \end{bmatrix}
\end{aligned}
\]

### Step 3: Solve the system from the echelon form.

The matrix is now:

\[
\begin{bmatrix}
2 & 1 & | & 8 \\
0 & -2.5 & | & -5
\end{bmatrix}
\]

From Row 2:

\[
-2.5y = -5 \quad \Rightarrow \quad y = 2
\]

Now substitute \(y = 2\) into Row 1:

\[
2x + y = 8 \quad \Rightarrow \quad 2x + 2 = 8 \quad \Rightarrow \quad 2x = 6 \quad \Rightarrow \quad x = 3
\]

### Step 4: Conclusion

The solution is \(x = 3\) and \(y = 2\), so the ordered pair is:

\[
\boxed{(3, 2)}
\]

Would you like any further details on this process?

### 5 Related Questions:
1. How do you perform row reduction to echelon form in a larger system of equations?
2. What is the significance of the echelon form in solving systems of equations?
3. Can Gaussian elimination be used for systems with more than two equations and unknowns?
4. What happens if there are infinitely many solutions in a system of equations?
5. How can systems of equations be solved using matrix inverses instead of Gaussian elimination?

### Tip:
Always verify your solution by substituting it back into the original system of equations to check for consistency.

Use the echelon method to solve the following system of two equations in two unknowns. Check your answer: 2x + y = 8, 3x - y = 7.

Learn how to solve a system of equations, 2x + y = 8 and 3x - y = 7, using the echelon method (Gaussian elimination). Follow a step-by-step solution to find the ordered pair (3, 2). This problem is ideal for students in Grades 8-10 and covers key algebraic methods.

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