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}
