If `(2 x - y)/( x + 2y ) = (1)/(2)` , then value of `(3 x - y)/( 3 x - y) ` is

Options

A. `(1)/(5)`
B. `(3)/(5)`
C. `(4)/(5)`
D. 1

Answer

To solve the equation \(\frac{2x - y}{x + 2y} = \frac{1}{2}\) and find the value of \(\frac{3x - y}{3x + y}\), we can follow these steps:

### Step 1: Cross-multiply the equation
Starting with the equation:
\[ \frac{2x - y}{x + 2y} = \frac{1}{2} \]
Cross-multiply to eliminate the fraction:
\[ 2(2x - y) = 1(x + 2y) \]

### Step 2: Distribute both sides
Distributing on both sides gives:
\[ 4x - 2y = x + 2y \]

### Step 3: Rearrange the equation
Now, let's move all terms involving \(x\) to one side and all terms involving \(y\) to the other side:
\[ 4x - x = 2y + 2y \]
This simplifies to:
\[ 3x = 4y \]

### Step 4: Express \(x\) in terms of \(y\)
Now, we can express \(x\) in terms of \(y\):
\[ x = \frac{4}{3}y \]

### Step 5: Substitute \(x\) into the expression \(\frac{3x - y}{3x + y}\)
Now we substitute \(x = \frac{4}{3}y\) into the expression \(\frac{3x - y}{3x + y}\):
\[ 3x = 3\left(\frac{4}{3}y\right) = 4y \]
Thus, we have:
\[ \frac{3x - y}{3x + y} = \frac{4y - y}{4y + y} = \frac{3y}{5y} \]

### Step 6: Simplify the expression
Now, simplify:
\[ \frac{3y}{5y} = \frac{3}{5} \]

### Final Answer
The value of \(\frac{3x - y}{3x + y}\) is:
\[ \frac{3}{5} \]

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