**Systems of two equations with two unknowns:**

Systems of two equations including two unknowns can also be solved through comparison.

**Step 1. **Solve each equation for the similar unknown in terms of the other unknown.

**Step 2. **Set the two expressions acquired equal to each other.

**Step 3.** Solve the resulting equation for the one remaining unknown.

**Step 4.** Find out the value of the other unknown through substituting the value of the first unknown into one of the original equations.

**Step 5.** Check the solution through substituting the values of the two unknowns into the other original equation.

**Example:**

Solve the subsequent system of equations by comparison.

5x + 6y = 12

3x + 5y = 3

**Solution:**

**Step 1.** Solve both equations for x.

5x +6y = 12

5x = 12 - 6y

5x/5 = (12-6y)/5

x = (12-6y)/5

3x + 5y = 3

3x = 3 - 5y

3x/3 = (3- 5y)/ 3

x = 3-5y /3

**Step 2.** Set the two values for x equal to each other.

(12-6y)/5 = (3-5y)/3

**Step 3.** Solve the resulting equation for y.

(12-6y)/ 5 = (3-5y) /3

(3)(5) (12-6y)/5 = (3-5y)/3 (3)(5)

3(12 - 6y) = 5 (3- 5y)

36- 18y = 15 - 25y

25y - 18y = 15 - 36

7y = -21

7y/ 7 = -21/7

y = -3

**Step 4. **Substitute y = -3 into one of the original equations and solve for x.

5x + 6y = 12

5x + 6(-3) = 12

5x -18 = 12

5x = 12 +18

5x = 30

5x/5 = 30/5

X = 6

**Step 5.** Check the solution by substituting x = 6 and y = -3 into the other original equation.

3x + 5y = 3

3(6) + 5 (-3) = 3

18- 15 =3

3 = 3

Therefore, the solution checks.

Quite often, when more than one unknown exists in a problem, the end result of the equations expressing the problem is a set of simultaneous equations showing the relationship of one of the unknowns to the other unknowns.