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

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 x:

   x + 3y = 7

   x = 7 - 3y

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

   2x - y = 5

   2(7 - 3y) - y = 5

3. Step 3: Simplify and solve for y.

   14 - 6y - y = 5

   14 - 7y = 5

   -7y = 5 - 14

   -7y = -9

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

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

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

   x = 7 - 3y

   x = 7 - 3(\(\frac{9}{7}\))

   x = 7 - \(\frac{27}{7}\)

   x = \(\frac{49}{7}\) - \(\frac{27}{7}\)

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

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