System of two linear equations in two variables:

Ax + By = C
Ex + Fy = G

The Substitution Method:

ex.
x + y = 2
2x - y = 7

1. Solve one of the equations for one variable in terms of the other,
try to avoid fractions.

x + y = 2
y = -x + 2

2. Substitute the expression above to the other equation.

2x - y = 7
2x - (-x + 2) = 7
2x + x - 2 = 7
3x - 2 = 7
3x = 9
x = 3

3. Substitute the solution above to one of the equations in the system
and solve.

x + y = 2
3 + y = 2
y = 2-3
y = -1

4. Check to see if the solutions obtained are right.

x + y = 2
3 - 1 = 2
2x - y = 7
2(3) - (-1) = 7
6 + 1 = 7

Elimination by Addition Method:

x + y = 2
2x - y = 7

1. Multiply one of the equation with an appropriate real number that will eliminate one of the variables when the equations are added together.
Multiply the first equation with 1 and add the first equation to the second.

2. Substitute x = 3 to one of the equation to find the value of y.

x = 3
x + y = 2
3 + y = 2
y = -1

3. Check the solutions with the equations.

x + y = 2
3 - 1 = 2
2x - y = 7
2(3) - (-1) = 7
6 + 1 = 7

System of Three Equations in Three Variables:

Ax + By +Cz = D
Ex + Fy +Gz = H
Ix + Jy +Kz = L

Solving system of three equations in three variables is similar to solving system with two variables. The solution set is a set of ordered triple of real numbers(x,y,z).

ex.

Now, add the first and third equations.

Add the two new solutions,

Substitute x = 5 into one of the two new solutions,

x + y = 1
5 + y = 1
y = -4

Substitute y = -4 and x = 5 into one of the original three equations,

x - 4y - z = 18
5 - 4(-4) - z = 18
5 + 16 - z = 18
5 + 16 - 18 = z
3 = z

The solution set is { 5,-4,3}