What are the value of x and y for the pair of linear equations `2x+3y=2` and `x-2y=8`?

Options

A. 4, 2
B. `-4, 2`
C. `4, -2`
D. `-4, -2`

Answer

To solve the pair of linear equations \(2x + 3y = 2\) and \(x - 2y = 8\), 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. \(2x + 3y = 2\) (Equation 1)
2. \(x - 2y = 8\) (Equation 2)

### Step 2: Make the coefficients of \(x\) equal
To eliminate \(x\), we can multiply Equation 2 by 2 so that the coefficient of \(x\) in both equations becomes equal.
\[ 2(x - 2y) = 2(8) \]
This gives us:
\[ 2x - 4y = 16 \quad (Equation 3) \]

### Step 3: Subtract the new equation from the first equation
Now we can subtract Equation 3 from Equation 1:
\[ (2x + 3y) - (2x - 4y) = 2 - 16 \]
This simplifies to:
\[ 2x + 3y - 2x + 4y = -14 \]
\[ 7y = -14 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 7:
\[ y = -2 \]

### Step 5: Substitute \(y\) back into one of the original equations
Now that we have \(y\), we can substitute it back into Equation 2 to find \(x\):
\[ x - 2(-2) = 8 \]
This simplifies to:
\[ x + 4 = 8 \]

### Step 6: Solve for \(x\)
Now, subtract 4 from both sides:
\[ x = 8 - 4 \]
\[ x = 4 \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = 4, \quad y = -2 \]

---