`2x+y=6`
`7x+2y=27`
The system of equations above is satisfied by which of the following ordered pairs `(x, y)`?

Options

A. `(-5, 4)`
B. `(4, -2)`
C. `(5, 4)`
D. `(5, -4)`

Answer

To solve the system of equations given by:

1. \( 2x + y = 6 \) (Equation 1)
2. \( 7x + 2y = 27 \) (Equation 2)

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

### Step 1: Make the coefficients of \( y \) the same in both equations.
To do this, we can multiply Equation 1 by 2:

\[ 2(2x + y) = 2(6) \]

This gives us:

\[ 4x + 2y = 12 \quad \text{(Equation 3)} \]

### Step 2: Write down the modified equations.
Now we have:

- Equation 3: \( 4x + 2y = 12 \)
- Equation 2: \( 7x + 2y = 27 \)

### Step 3: Subtract Equation 3 from Equation 2.
Now we will subtract Equation 3 from Equation 2 to eliminate \( y \):

\[ (7x + 2y) - (4x + 2y) = 27 - 12 \]

This simplifies to:

\[ 7x - 4x + 2y - 2y = 15 \]

Which further simplifies to:

\[ 3x = 15 \]

### Step 4: Solve for \( x \).
Now, divide both sides by 3:

\[ x = \frac{15}{3} = 5 \]

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

\[ 2(5) + y = 6 \]

This gives:

\[ 10 + y = 6 \]

### Step 6: Solve for \( y \).
Now, isolate \( y \):

\[ y = 6 - 10 = -4 \]

### Final Answer:
Thus, the solution to the system of equations is:

\[ (x, y) = (5, -4) \]