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.