Which of the following (x,y) pairs is the solution for the system of equations `x + 2y = 2 and -2x + y = 16`?

Options

A. `(-6, 4)`
B. `(-1, 1, 5)`
C. `(1,0,5)`
D. `(0,1)`

Answer

To solve the system of equations given by \( x + 2y = 2 \) and \( -2x + y = 16 \), we can follow these steps:

### Step 1: Write down the equations
We have the two equations:
1. \( x + 2y = 2 \) (Equation 1)
2. \( -2x + y = 16 \) (Equation 2)

### Step 2: Solve for \( x \) in terms of \( y \)
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 2 - 2y \]

### Step 3: Substitute \( x \) in Equation 2
Now, we will substitute the expression for \( x \) into Equation 2:
\[ -2(2 - 2y) + y = 16 \]

### Step 4: Simplify the equation
Distributing the -2 gives:
\[ -4 + 4y + y = 16 \]
Combining like terms:
\[ -4 + 5y = 16 \]

### Step 5: Solve for \( y \)
Now, add 4 to both sides:
\[ 5y = 20 \]
Now, divide by 5:
\[ y = 4 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into the equation for \( x \):
\[ x = 2 - 2(4) = 2 - 8 = -6 \]

### Step 7: Write the solution pair
Thus, the solution pair \( (x, y) \) is:
\[ (-6, 4) \]

### Final Answer
The solution for the system of equations is \( (-6, 4) \).

---