If 2x + 3y + z = 55, x+ z- y = 4 and y – x + z = 12, what is value of y?

Options

A. 7
B. 8
C. 12
D. None of these

Answer

To solve the system of equations given:

1. \(2x + 3y + z = 55\) (Equation 1)
2. \(x + z - y = 4\) (Equation 2)
3. \(y - x + z = 12\) (Equation 3)

We will follow these steps:

### Step 1: Add Equation 2 and Equation 3
First, we will add Equation 2 and Equation 3 to eliminate \(x\) and \(y\):

\[ (x + z - y) + (y - x + z) = 4 + 12 \]

This simplifies to:

\[ 2z = 16 \]

### Step 2: Solve for \(z\)
Now, we can solve for \(z\):

\[ z = \frac{16}{2} = 8 \]

### Step 3: Substitute \(z\) into Equation 2
Next, we substitute \(z = 8\) back into Equation 2:

\[ x + 8 - y = 4 \]

This simplifies to:

\[ x - y = 4 - 8 \]
\[ x - y = -4 \quad \text{(Equation 4)} \]

### Step 4: Substitute \(z\) into Equation 1
Now, we substitute \(z = 8\) into Equation 1:

\[ 2x + 3y + 8 = 55 \]

This simplifies to:

\[ 2x + 3y = 55 - 8 \]
\[ 2x + 3y = 47 \quad \text{(Equation 5)} \]

### Step 5: Solve Equations 4 and 5
Now we have two equations (Equation 4 and Equation 5):

1. \(x - y = -4\) (Equation 4)
2. \(2x + 3y = 47\) (Equation 5)

From Equation 4, we can express \(x\) in terms of \(y\):

\[ x = y - 4 \]

### Step 6: Substitute \(x\) into Equation 5
Now, substitute \(x\) into Equation 5:

\[ 2(y - 4) + 3y = 47 \]

This simplifies to:

\[ 2y - 8 + 3y = 47 \]
\[ 5y - 8 = 47 \]

### Step 7: Solve for \(y\)
Now, we can solve for \(y\):

\[ 5y = 47 + 8 \]
\[ 5y = 55 \]
\[ y = \frac{55}{5} = 11 \]

### Final Answer
The value of \(y\) is \(11\).

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