Solve the system of equations using the elimination method: 3x + 2y = 11 5x - 2y = 9

Method: Elimination

The elimination method involves manipulating one or both equations so that the coefficients of one variable are opposites. Then, adding the equations together eliminates that variable, allowing us to solve for the remaining one.

Step-by-step solution:

1. Step 1: Observe the coefficients of the variables. In the given system:

   3x + 2y = 11

   5x - 2y = 9

   The coefficients of 'y' are +2 and -2, which are opposites. This means we can directly add the equations.
2. Step 2: Add the two equations together.

   (3x + 2y) + (5x - 2y) = 11 + 9

   3x + 5x + 2y - 2y = 20

   8x = 20

3. Step 3: Solve for x.

   8x = 20

   x = \(\frac{20}{8}\)

   x = \(\frac{5}{2}\)

4. Step 4: Substitute the value of x into either of the original equations to solve for y. Let's use the first equation.

   3x + 2y = 11

   3(\(\frac{5}{2}\)) + 2y = 11

   \(\frac{15}{2}\) + 2y = 11

5. Step 5: Solve for y.

   2y = 11 - \(\frac{15}{2}\)

   2y = \(\frac{22}{2}\) - \(\frac{15}{2}\)

   2y = \(\frac{7}{2}\)

   y = \(\frac{7}{4}\)

Answer: x = \(\frac{5}{2}\), y = \(\frac{7}{4}\)