If ` 6 x - 5y = 13 , 7 x + 2y = 23 ` then 11 x + 18 y`

Options

A. `-15`
B. 51
C. 33
D. 15

Answer

To solve the given equations \(6x - 5y = 13\) and \(7x + 2y = 23\) and find the value of \(11x + 18y\), we can follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(6x - 5y = 13\) (Equation 1)
2. \(7x + 2y = 23\) (Equation 2)

### Step 2: Make the coefficients of \(y\) the same
To eliminate \(y\), we can multiply Equation 1 by \(2\) and Equation 2 by \(5\) to make the coefficients of \(y\) equal:

- Multiply Equation 1 by \(2\):
\[ 2(6x - 5y) = 2(13) \implies 12x - 10y = 26 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by \(5\):
\[ 5(7x + 2y) = 5(23) \implies 35x + 10y = 115 \quad \text{(Equation 4)} \]

### Step 3: Add the two new equations
Now we can add Equation 3 and Equation 4:
\[ (12x - 10y) + (35x + 10y) = 26 + 115 \]
This simplifies to:
\[ 12x + 35x = 141 \]
\[ 47x = 141 \]

### Step 4: Solve for \(x\)
Now, divide both sides by \(47\):
\[ x = \frac{141}{47} = 3 \]

### Step 5: Substitute \(x\) back to find \(y\)
Now that we have \(x\), we can substitute it back into either of the original equations to find \(y\). We will use Equation 2:
\[ 7x + 2y = 23 \]
Substituting \(x = 3\):
\[ 7(3) + 2y = 23 \]
\[ 21 + 2y = 23 \]
Subtract \(21\) from both sides:
\[ 2y = 2 \]
Now divide by \(2\):
\[ y = 1 \]

### Step 6: Calculate \(11x + 18y\)
Now we can find the value of \(11x + 18y\):
\[ 11x + 18y = 11(3) + 18(1) \]
Calculating this gives:
\[ = 33 + 18 = 51 \]

### Final Answer
Thus, the value of \(11x + 18y\) is \(51\).

---