`2c+3d=17`
`6c+5d=39`
In the system of linear equations above. What is the value of 4c-4d?

Options

A. -4
B. 1
C. 4
D. 13

Answer

To solve the system of equations given by:

1. \( 2c + 3d = 17 \) (Equation 1)
2. \( 6c + 5d = 39 \) (Equation 2)

we want to find the value of \( 4c - 4d \).

### Step 1: Multiply Equation 1 by 3
To eliminate \( c \) from the equations, we can multiply Equation 1 by 3:

\[ 3(2c + 3d) = 3(17) \]

This gives us:

\[ 6c + 9d = 51 \quad \text{(Equation 3)} \]

### Step 2: Subtract Equation 2 from Equation 3
Now, we will subtract Equation 2 from Equation 3:

\[ (6c + 9d) - (6c + 5d) = 51 - 39 \]

This simplifies to:

\[ 4d = 12 \]

### Step 3: Solve for \( d \)
Now, divide both sides by 4 to find \( d \):

\[ d = \frac{12}{4} = 3 \]

### Step 4: Substitute \( d \) back into Equation 1
Next, we substitute \( d = 3 \) back into Equation 1 to find \( c \):

\[ 2c + 3(3) = 17 \]

This simplifies to:

\[ 2c + 9 = 17 \]

### Step 5: Solve for \( c \)
Now, isolate \( c \):

\[ 2c = 17 - 9 \]
\[ 2c = 8 \]
\[ c = \frac{8}{2} = 4 \]

### Step 6: Calculate \( 4c - 4d \)
Now that we have \( c = 4 \) and \( d = 3 \), we can calculate \( 4c - 4d \):

\[ 4c - 4d = 4(4) - 4(3) \]
\[ = 16 - 12 \]
\[ = 4 \]

Thus, the value of \( 4c - 4d \) is \( 4 \).

### Final Answer
The value of \( 4c - 4d \) is \( 4 \).

---