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

Method: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This allows us to solve for the remaining variable.

Step-by-step solution:

1. Step 1: Choose an equation and solve for one variable. From the first equation, we can solve for y:

   3x + 2y = 11

   2y = 11 - 3x

   y = \(\frac{11 - 3x}{2}\)

2. Step 2: Substitute the expression for y into the second equation.

   5x - 2y = 9

   5x - 2(\(\frac{11 - 3x}{2}\)) = 9

3. Step 3: Simplify and solve for x.

   5x - (11 - 3x) = 9

   5x - 11 + 3x = 9

   8x - 11 = 9

   8x = 9 + 11

   8x = 20

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

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

4. Step 4: Substitute the value of x back into the equation for y.

   y = \(\frac{11 - 3x}{2}\)

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

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

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

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

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

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