Solve using augmented matrices.




2

2

x 1

x1

plus

+

3 x 2 equals

3x2=

11

11

x 1

x1

minus

−

x 2

x2

equals

=

negative 2

−2







**A.** The unique solution is x1=enter your response here and x2=enter your response here.




**B.** The system has infinitely many solutions. The solution is x1=enter your response here and x2=t.

​(Simplify your answer. Type an expression using t as the​ variable.)




**C.** There is no solution.

We are given the system of linear equations:

\[
\begin{aligned}
1. \quad 2x_1 + 3x_2 &= 11 \\
2. \quad x_1 - x_2 &= -2
\end{aligned}
\]

We can solve this system using augmented matrices.

### Step 1: Write the augmented matrix
The augmented matrix for this system is:

\[
\begin{pmatrix}
2 & 3 & | & 11 \\
1 & -1 & | & -2
\end{pmatrix}
\]

### Step 2: Perform row operations to reduce the matrix

**Operation 1:** Subtract \(\frac{1}{2}\) of row 1 from row 2 to eliminate \(x_1\) in the second row.

\[
R_2 \to R_2 - \frac{1}{2} R_1
\]

\[
\begin{pmatrix}
2 & 3 & | & 11 \\
1 & -1 & | & -2
\end{pmatrix}
\quad \Rightarrow \quad
\begin{pmatrix}
2 & 3 & | & 11 \\
0 & -\frac{7}{2} & | & -\frac{15}{2}
\end{pmatrix}
\]

**Operation 2:** Multiply row 2 by \(-\frac{2}{7}\) to simplify the second row.

\[
R_2 \to -\frac{2}{7} R_2
\]

\[
\begin{pmatrix}
2 & 3 & | & 11 \\
0 & 1 & | & \frac{15}{7}
\end{pmatrix}
\]

**Operation 3:** Subtract \(3\) times row 2 from row 1 to eliminate \(x_2\) in the first row.

\[
R_1 \to R_1 - 3 R_2
\]

\[
\begin{pmatrix}
2 & 0 & | & \frac{8}{7} \\
0 & 1 & | & \frac{15}{7}
\end{pmatrix}
\]

### Step 3: Solve for \(x_1\) and \(x_2\)

From the matrix, we have:
\[
2x_1 = \frac{8}{7} \quad \Rightarrow \quad x_1 = \frac{4}{7}
\]
\[
x_2 = \frac{15}{7}
\]

Thus, the unique solution is:
\[
x_1 = \frac{4}{7}, \quad x_2 = \frac{15}{7}
\]

### Answer:
**A.** The unique solution is \(x_1 = \frac{4}{7}\) and \(x_2 = \frac{15}{7}\).

---

Would you like more details on any step?

Here are 5 related questions you might explore next:
1. How do you solve systems of equations using Gaussian elimination?
2. What are the conditions for a system to have infinitely many solutions?
3. How do you represent a system of equations as an augmented matrix?
4. Can augmented matrices be used for non-linear systems of equations?
5. What is the difference between row echelon form and reduced row echelon form?

**Tip:** Always check your row operations carefully when solving with matrices to avoid arithmetic errors!

Solve using augmented matrices: 2x1 + 3x2 = 11, x1 - x2 = -2.

Learn how to solve a system of linear equations using augmented matrices with a step-by-step solution. This problem involves matrix row operations and Gaussian elimination to find the unique solution. Suitable for high school students in Grades 10-12.

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