`3a-7b=4d-9`
`-4c+10a=6b+7`
`-2a+3c-4d=10`
Given the system of equation above, what is the value of `-10a-2b+2c`?

Options

A. `-52`
B. `-26`
C. `8`
D. `26`

Answer

To solve the given system of equations and find the value of \(-10a - 2b + 2c\), we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \(3a - 7b = 4d - 9\)
2. \(-4c + 10a = 6b + 7\)
3. \(-2a + 3c - 4d = 10\)

### Step 2: Rearrange equations to isolate \(d\)
From equation 1, we can express \(d\):
\[ 4d = 3a - 7b + 9 \quad \text{(Equation 1)} \]
From equation 3, we can also express \(d\):
\[ 4d = -2a + 3c - 10 \quad \text{(Equation 3)} \]

### Step 3: Set the two expressions for \(d\) equal to each other
Since both expressions equal \(4d\), we can set them equal:
\[ 3a - 7b + 9 = -2a + 3c - 10 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 3a + 2a - 7b - 3c + 9 + 10 = 0 \]
This simplifies to:
\[ 5a - 7b - 3c = -19 \quad \text{(Equation 4)} \]

### Step 5: Rearrange equation 2
From equation 2, we can rearrange it as follows:
\[ 10a - 6b - 4c = 7 \]

### Step 6: Subtract equation 4 from equation 2
Now, we will subtract equation 4 from the rearranged equation 2:
\[ (10a - 6b - 4c) - (5a - 7b - 3c) = 7 - (-19) \]
This simplifies to:
\[ 10a - 5a - 6b + 7b - 4c + 3c = 7 + 19 \]
Which simplifies to:
\[ 5a + b - c = 26 \quad \text{(Equation 5)} \]

### Step 7: Relate to the required expression
We need to find the value of \(-10a - 2b + 2c\). Notice that if we multiply equation 5 by \(-2\), we will get:
\[ -2(5a + b - c) = -2 \cdot 26 \]
This gives:
\[ -10a - 2b + 2c = -52 \]

### Final Answer
Thus, the value of \(-10a - 2b + 2c\) is:
\[ \boxed{-52} \]