2x +3y =12

To solve the system of equations using the elimination method, we'll start with the two equations:

1. \( 2x + 3y = 12 \) (Equation 1)
2. \( x + 4y = 11 \) (Equation 2)

The goal of the elimination method is to eliminate one of the variables by adding or subtracting the equations after multiplying them by suitable coefficients.

Step 1: Make the coefficients of x or y the same

We can eliminate \( x \) by making the coefficients of \( x \) the same in both equations. To do this, we can multiply Equation 2 by 2:

\[ 2(x + 4y) = 2(11) \]

This gives us:

\[ 2x + 8y = 22 \quad \text{(Equation 3)} \]

Step 2: Subtract the equations

Now we have:

1. \( 2x + 3y = 12 \) (Equation 1)
2. \( 2x + 8y = 22 \) (Equation 3)

Now, we can subtract Equation 1 from Equation 3:

\[ (2x + 8y) - (2x + 3y) = 22 - 12 \] \[ 2x + 8y - 2x - 3y = 10 \] \[ 5y = 10 \]

Step 3: Solve for y

Dividing both sides by 5:

\[ y = 2 \]

Step 4: Substitute back to find x

Now that we have \( y \), we can substitute \( y = 2 \) back into one of the original equations to find \( x \). We'll use Equation 2:

\[ x + 4(2) = 11 \] \[ x + 8 = 11 \] \[ x = 11 - 8 \] \[ x = 3 \]

Final solution

The solution to the system of equations is:

\[ x = 3, \quad y = 2 \]

So, the ordered pair is \( (3, 2) \).