If `2x + 3y =0 and 3x - 4y = 34,` then `x -y =`____

Options

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

Answer

To solve the equations \(2x + 3y = 0\) and \(3x - 4y = 34\) and find the value of \(x - y\), we can follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \(2x + 3y = 0\) (Equation 1)
2. \(3x - 4y = 34\) (Equation 2)

### Step 2: Eliminate one variable
To eliminate \(x\), we can multiply Equation 1 by 3 and Equation 2 by 2:
- Multiply Equation 1 by 3:
\[ 3(2x + 3y) = 3(0) \implies 6x + 9y = 0 \quad (Equation 3) \]
- Multiply Equation 2 by 2:
\[ 2(3x - 4y) = 2(34) \implies 6x - 8y = 68 \quad (Equation 4) \]

### Step 3: Subtract the equations
Now, we can subtract Equation 4 from Equation 3 to eliminate \(x\):
\[ (6x + 9y) - (6x - 8y) = 0 - 68 \]
This simplifies to:
\[ 9y + 8y = -68 \implies 17y = -68 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 17:
\[ y = \frac{-68}{17} = -4 \]

### Step 5: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute it back into Equation 1 to find \(x\):
\[ 2x + 3(-4) = 0 \]
This simplifies to:
\[ 2x - 12 = 0 \implies 2x = 12 \implies x = \frac{12}{2} = 6 \]

### Step 6: Find \(x - y\)
Now we can find \(x - y\):
\[ x - y = 6 - (-4) = 6 + 4 = 10 \]

### Final Answer
Thus, the value of \(x - y\) is \(10\).

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