Solve the simultaneous equations: `5x-3y = 8, 3x+ y =2`

Answer

To solve the simultaneous equations \(5x - 3y = 8\) and \(3x + y = 2\), we will use the elimination method. Here’s a step-by-step solution:

### Step 1: Write down the equations
We have the following two equations:
1. \(5x - 3y = 8\) (Equation 1)
2. \(3x + y = 2\) (Equation 2)

### Step 2: Make the coefficients of \(y\) the same
To eliminate \(y\), we can manipulate the second equation. We will multiply Equation 2 by 3 to match the coefficient of \(y\) in Equation 1.

\[ 3(3x + y) = 3(2) \]
This gives us:
\[ 9x + 3y = 6 \quad \text{(Equation 3)} \]

### Step 3: Add the two equations
Now we will add Equation 1 and Equation 3 together:

\[ (5x - 3y) + (9x + 3y) = 8 + 6 \]

This simplifies to:
\[ 5x + 9x - 3y + 3y = 14 \]
\[ 14x = 14 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):

\[ x = \frac{14}{14} = 1 \]

### Step 5: Substitute \(x\) back into one of the original equations
We will substitute \(x = 1\) back into Equation 2 to find \(y\):

\[ 3(1) + y = 2 \]
\[ 3 + y = 2 \]

### Step 6: Solve for \(y\)
Now, we can solve for \(y\):

\[ y = 2 - 3 \]
\[ y = -1 \]

### Final Answer
The solution to the simultaneous equations is:
\[ x = 1, \quad y = -1 \]