Solve for x and y:
`4x-2y=3`, `2x+y=1`

Answer

To solve the system of equations given by:

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

we will use the elimination method. Here are the steps:

### Step 1: Manipulate the equations to align coefficients

We can manipulate Equation 2 to make it easier to eliminate \(y\). We can multiply Equation 2 by 2 to align the coefficients of \(y\):

\[ 2(2x + y) = 2(1) \]

This gives us:

\[ 4x + 2y = 2 \quad \text{(Equation 3)} \]

### Step 2: Add the modified Equation 3 to Equation 1

Now we can add Equation 1 and Equation 3:

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

This simplifies to:

\[ 8x = 5 \]

### Step 3: Solve for \(x\)

Now, we can solve for \(x\):

\[ x = \frac{5}{8} \]

### Step 4: Substitute \(x\) back into one of the original equations to find \(y\)

We will substitute \(x = \frac{5}{8}\) back into Equation 2:

\[ 2\left(\frac{5}{8}\right) + y = 1 \]

This simplifies to:

\[ \frac{10}{8} + y = 1 \]

### Step 5: Solve for \(y\)

Now, we can isolate \(y\):

\[ y = 1 - \frac{10}{8} \]

This can be rewritten as:

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

To combine the fractions, we convert 1 into a fraction with a denominator of 4:

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

### Final Solution

Thus, the solution to the system of equations is:

\[ x = \frac{5}{8}, \quad y = \frac{-1}{4} \]

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