Let `2x + 3y = 4` and `5x + 6y = 7`. What is the value of `8x + 9y` ?

Options

A. `-10`
B. `-1`
C. `2`
D. 10

Answer

To find the value of \( 8x + 9y \) given the equations \( 2x + 3y = 4 \) and \( 5x + 6y = 7 \), we can follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( 2x + 3y = 4 \) (Equation 1)
2. \( 5x + 6y = 7 \) (Equation 2)

### Step 2: Multiply Equation 1 by 2
To eliminate \( y \), we can manipulate the equations. Let's multiply Equation 1 by 2:
\[ 2(2x + 3y) = 2 \cdot 4 \]
This gives us:
\[ 4x + 6y = 8 \quad \text{(Equation 3)} \]

### Step 3: Subtract Equation 2 from Equation 3
Now, we will subtract Equation 2 from Equation 3:
\[ (4x + 6y) - (5x + 6y) = 8 - 7 \]
This simplifies to:
\[ 4x + 6y - 5x - 6y = 1 \]
Which further simplifies to:
\[ -x = 1 \]

### Step 4: Solve for \( x \)
From the equation \( -x = 1 \), we find:
\[ x = -1 \]

### Step 5: Substitute \( x \) back into Equation 1 to find \( y \)
Now we substitute \( x = -1 \) back into Equation 1:
\[ 2(-1) + 3y = 4 \]
This simplifies to:
\[ -2 + 3y = 4 \]
Adding 2 to both sides gives:
\[ 3y = 6 \]
Now, dividing by 3:
\[ y = 2 \]

### Step 6: Substitute \( x \) and \( y \) into \( 8x + 9y \)
Now we can find \( 8x + 9y \):
\[ 8(-1) + 9(2) = -8 + 18 \]
This simplifies to:
\[ -8 + 18 = 10 \]

### Final Answer
Thus, the value of \( 8x + 9y \) is \( \boxed{10} \).