Systems of Linear
Equations
I
System of
Equations
Related
topics:1, 9, 10.
A system of two linear equations in two variables is called the system formed by two equations:
Ax + By = C
Ex + Fy = G
There are two main techniques for solving this kind of systems.
I. The Substitution Method
Consider the following system:
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
II. 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.
x + y =
2
2x - y = 7
3x =9
Solve 3x=9 ; get x=3.
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).
Example:
2x + 3y + z = 1
5x + 2y - 3z= 8
x - 4y
- z = 18
Multiply the first equation by 3 and add the result to the second equations:
3(2x +
3y + z) = 3
5x + 2y - 3z =
8
11x +
11y = 11
factor 11 out and
get
x + y =1
Now, add the first and third equations.
2x + 3y + z = 1
x - 4y - z = 18
3x -
y
=19
Add the two new solutions,
x + y =
1
3x - y = 19
4x
=20
x = 5
Substitute x = 5 into one of the two new equations, to obtain y:
x + y =
1
5 + y = 1
y =
-4
Substitute y = -4 and x = 5 into one of the original three equations to get z:
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}.