Question

Two systems of equations are given below. For each system, choose the best description of its solution. applicable, give the solution. If ? 00 System A 3x-y=-8 -3x+y=8 The system has no solution. The system has a unique solution: (x,y) =) The system has infinitely many solutions. The system has no solution. The system has a unique solution: System B x+3y=11 (x,y) = () The system has infinitely many solutions. X 5 Submit Assignment -x+3y=7

Answer 1

To analyze both systems of equations given, we'll identify whether they have a unique solution, infinitely many solutions, or no solution.

### System A:

1. **Equations**:
\[
3x - y = -8 \quad (1)
\]
\[
-3x + y = 8 \quad (2)
\]

2. **Rearranging the equations** for comparison:
- From equation (1): \( y = 3x + 8 \)
- From equation (2): \( y = 3x + 8 \)

3. **Analysis**:
The two equations are identical, which means they represent the same line. Thus, there are infinitely many solutions.

**Conclusion for System A**: The system has infinitely many solutions.

### System B:

1. **Equations**:
\[
x + 3y = 11 \quad (3)
\]
\[
-x + 3y = 7 \quad (4)
\]

2. **Rearranging the equations**:
- From equation (3): \( 3y = 11 - x \) or \( y = \frac{11 - x}{3} \)
- From equation (4): \( 3y = 7 + x \) or \( y = \frac{7 + x}{3} \)

3. **Setting the equations for \(y\) equal to find \(x\)**:
\[
\frac{11 - x}{3} = \frac{7 + x}{3}
\]

4. **Multiplying both sides by 3 to eliminate the denominator**:
\[
11 - x = 7 + x
\]

5. **Solving for \(x\)**:
\[
11 - 7 = 2x \\
4 = 2x \\
x = 2
\]

6. **Substituting \(x = 2\) back into one of the original equations to find \(y\)**:
Using equation (3):
\[
2 + 3y = 11 \quad \Rightarrow \quad 3y = 9 \quad \Rightarrow \quad y = 3
\]

**Conclusion for System B**: The system has a unique solution: \((x, y) = (2, 3)\).

### Final Descriptions:

- **System A**: The system has infinitely many solutions.
- **System B**: The system has a unique solution: \((x, y) = (2, 3)\).