Solve the system of equations using the elimination method: x + 3y = 7 2x - y = 5

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: Manipulate the equations to have opposite coefficients for one variable. We can multiply the second equation by 3 to make the coefficients of 'y' opposites:

   Equation 1: x + 3y = 7

   Equation 2 (multiplied by 3): 3 * (2x - y) = 3 * 5

   6x - 3y = 15

2. Step 2: Add the modified equations together.

   (x + 3y) + (6x - 3y) = 7 + 15

   x + 6x + 3y - 3y = 22

   7x = 22

3. Step 3: Solve for x.

   7x = 22

   x = \(\frac{22}{7}\)

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

   x + 3y = 7

   \(\frac{22}{7}\) + 3y = 7

5. Step 5: Solve for y.

   3y = 7 - \(\frac{22}{7}\)

   3y = \(\frac{49}{7}\) - \(\frac{22}{7}\)

   3y = \(\frac{27}{7}\)

   y = \(\frac{27}{7 imes 3}\)

   y = \(\frac{27}{21}\)

   y = \(\frac{9}{7}\)

Answer: x = \(\frac{22}{7}\), y = \(\frac{9}{7}\)