Question

Solve the system of equations.
3x - 4y = 40
5x - 5y = 55

A. (18, 7)
B. (4, -7)
C. (14, 3)
D. (8, -3)

How many solutions does the system have?
12x - 15y = 18
4x - 5y = 6
A. Two solutions
B. Infinitely many solutions
C. One solution
D. No solution

Solve the system of two linear equations.
-3x-4y=-23
3x-5y=5

A. (-5, -2)
B. (5, -2)
C. (-5, 2)
D. (5, 2)

Answer 1

Let's solve each of the systems of equations step by step.

### System 1:
1. **Equations:**
\[
3x - 4y = 40 \quad \text{(1)}
\]
\[
5x - 5y = 55 \quad \text{(2)}
\]

2. **Rearranging (2):**
Divide equation (2) by 5:
\[
x - y = 11 \quad \text{(3)}
\]
Then, \(y = x - 11\).

3. **Substituting (3) into (1):**
\[
3x - 4(x - 11) = 40
\]
\[
3x - 4x + 44 = 40
\]
\[
-x + 44 = 40
\]
\[
-x = 40 - 44 = -4 \Rightarrow x = 4.
\]

4. **Finding y:**
\[
y = 4 - 11 = -7.
\]

5. **Solution:**
\( (x, y) = (4, -7) \).
**The answer is B. (4, -7)**.

### For the second system:
1. **Equations:**
\[
12x - 15y = 18 \quad \text{(4)}
\]
\[
4x - 5y = 6 \quad \text{(5)}
\]

2. **Rearranging (5):**
Multiply (5) by 3:
\[
12x - 15y = 18 \quad \text{(6)}
\]

Notice that equation (6) is the same as equation (4).

3. **Conclusion:**
Both equations represent the same line, so they have infinitely many solutions.

**The answer is B. Infinitely many solutions**.

### System 3:
1. **Equations:**
\[
-3x-4y=-23 \quad \text{(7)}
\]
\[
3x-5y=5 \quad \text{(8)}
\]

2. **Adding equations (7) and (8):**
\[
-3x - 4y + 3x - 5y = -23 + 5
\]
\[
-9y = -18 \Rightarrow y = 2.
\]

3. **Finding x using (8):**
Substitute \( y = 2 \) back into (8):
\[
3x - 5(2) = 5
\]
\[
3x - 10 = 5
\]
\[
3x = 15 \Rightarrow x = 5.
\]

4. **Solution:**
\( (x, y) = (5, 2) \).
**The answer is D. (5, 2)**.

### Summary of Answers:
1. **First System:** B. (4, -7)
2. **Second System:** B. Infinitely many solutions
3. **Third System:** D. (5, 2)