Solve for x and y
`(3x-y)/(5)=2y-1, (3x)/(8)-(y)/(4)=(1)/(2)`

Answer

To solve the system of equations given by:

1. \(\frac{3x - y}{5} = 2y - 1\)
2. \(\frac{3x}{8} - \frac{y}{4} = \frac{1}{2}\)

we will follow these steps:

### Step 1: Simplify the first equation
Start with the first equation:

\[ \frac{3x - y}{5} = 2y - 1 \]

Cross-multiply to eliminate the fraction:

\[ 3x - y = 5(2y - 1) \]

Distributing the 5 on the right side:

\[ 3x - y = 10y - 5 \]

Now, rearranging the equation to isolate \(3x\):

\[ 3x = 10y - 5 + y \]

Combine like terms:

\[ 3x = 11y - 5 \]

Now, divide everything by 3 to express \(x\) in terms of \(y\):

\[ x = \frac{11y - 5}{3} \]

### Step 2: Substitute \(x\) in the second equation
Now, substitute \(x\) into the second equation:

\[ \frac{3x}{8} - \frac{y}{4} = \frac{1}{2} \]

Substituting \(x\):

\[ \frac{3\left(\frac{11y - 5}{3}\right)}{8} - \frac{y}{4} = \frac{1}{2} \]

The \(3\) in the numerator and denominator cancels out:

\[ \frac{11y - 5}{8} - \frac{y}{4} = \frac{1}{2} \]

### Step 3: Find a common denominator
The common denominator for \(8\) and \(4\) is \(8\). Rewrite \(\frac{y}{4}\) with a denominator of \(8\):

\[ \frac{11y - 5}{8} - \frac{2y}{8} = \frac{1}{2} \]

Combine the fractions:

\[ \frac{11y - 5 - 2y}{8} = \frac{1}{2} \]

This simplifies to:

\[ \frac{9y - 5}{8} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for \(y\)
Cross-multiply to eliminate the fraction:

\[ 9y - 5 = 4 \]

Now, add \(5\) to both sides:

\[ 9y = 4 + 5 \]

This simplifies to:

\[ 9y = 9 \]

Now, divide by \(9\):

\[ y = 1 \]

### Step 5: Substitute \(y\) back to find \(x\)
Now that we have \(y\), substitute it back into the equation for \(x\):

\[ x = \frac{11(1) - 5}{3} \]

This simplifies to:

\[ x = \frac{11 - 5}{3} = \frac{6}{3} = 2 \]

### Final Solution
Thus, the solution for the system of equations is:

\[ x = 2, \quad y = 1 \]