`3c-4d=-11`
`4c-3d=-3`
If `(c, d)` is a solution of the system of equations above, what is the value of c-d?

Options

A. `8`
B. `-2`
C. `-8`
D. `-14`

Answer

To solve the system of equations:

1. **Equations Given:**
\[ 3c - 4d = -11 \quad (1) \]
\[ 4c - 3d = -3 \quad (2) \]

2. **Multiply the Equations:**
We will use the elimination method to eliminate one variable. To make the coefficients of \(c\) the same in both equations, we can multiply the first equation by 4 and the second equation by 3.
\[ 4 \times (3c - 4d) = 4 \times (-11) \implies 12c - 16d = -44 \quad (3) \]
\[ 3 \times (4c - 3d) = 3 \times (-3) \implies 12c - 9d = -9 \quad (4) \]

3. **Subtract the Equations:**
Now, we will subtract equation (4) from equation (3):
\[ (12c - 16d) - (12c - 9d) = -44 - (-9) \]
Simplifying this gives:
\[ -16d + 9d = -44 + 9 \]
\[ -7d = -35 \]

4. **Solve for \(d\):**
Dividing both sides by -7:
\[ d = \frac{-35}{-7} = 5 \]

5. **Substitute \(d\) back into one of the original equations:**
We can substitute \(d = 5\) into equation (2):
\[ 4c - 3(5) = -3 \]
Simplifying this gives:
\[ 4c - 15 = -3 \]
Adding 15 to both sides:
\[ 4c = 12 \]

6. **Solve for \(c\):**
Dividing both sides by 4:
\[ c = \frac{12}{4} = 3 \]

7. **Find \(c - d\):**
Now we need to find \(c - d\):
\[ c - d = 3 - 5 = -2 \]

**Final Answer:**
The value of \(c - d\) is \(-2\).

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