If ` 3 x + 4y - 11= 18 and 8x - 6y + 12 = 6 ` , then what is the value of ` 5x - 3y - 9` ?

Options

A. 18
B. `-9`
C. `-27`
D. `-18`

Answer

To solve the equations and find the value of \( 5x - 3y - 9 \), we will follow these steps:

### Step 1: Rewrite the equations
We start with the two equations given:
1. \( 3x + 4y - 11 = 18 \)
2. \( 8x - 6y + 12 = 6 \)

We can rewrite them in a more standard form:
1. \( 3x + 4y = 29 \) (by adding 11 to both sides)
2. \( 8x - 6y = -6 \) (by subtracting 12 from both sides)

### Step 2: Eliminate one variable
To eliminate one variable, we can manipulate the equations. We can multiply the first equation by 2 and the second equation by 1 to align the coefficients of \( x \):
1. \( 2(3x + 4y) = 2(29) \) → \( 6x + 8y = 58 \)
2. \( 8x - 6y = -6 \)

### Step 3: Solve the system of equations
Now we have:
1. \( 6x + 8y = 58 \)
2. \( 8x - 6y = -6 \)

Next, we can multiply the first equation by 3 and the second by 4 to eliminate \( y \):
1. \( 3(6x + 8y) = 3(58) \) → \( 18x + 24y = 174 \)
2. \( 4(8x - 6y) = 4(-6) \) → \( 32x - 24y = -24 \)

Now we can add these two equations:
\[ (18x + 24y) + (32x - 24y) = 174 - 24 \]
This simplifies to:
\[ 50x = 150 \]
Thus, we find:
\[ x = \frac{150}{50} = 3 \]

### Step 4: Substitute to find \( y \)
Now that we have \( x = 3 \), we can substitute this value back into one of the original equations to find \( y \). We can use the first equation:
\[ 3(3) + 4y = 29 \]
This simplifies to:
\[ 9 + 4y = 29 \]
Subtracting 9 from both sides gives:
\[ 4y = 20 \]
Thus:
\[ y = \frac{20}{4} = 5 \]

### Step 5: Find the value of \( 5x - 3y - 9 \)
Now we substitute \( x = 3 \) and \( y = 5 \) into \( 5x - 3y - 9 \):
\[ 5(3) - 3(5) - 9 = 15 - 15 - 9 = -9 \]

### Final Answer
The value of \( 5x - 3y - 9 \) is \( -9 \).

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