Solve 3 linear equations (Algebra 2 using substitution method)

In the past articles, I have shown the readers how to solve 2 linear equations using substitution method and linear combination method.  In the next few articles, I will show you that the same methods can be used to solve 3 linear equations also.  The methods do not change, but it is just "tedious" work.

In this article, we will solve 3 linear equations using the substitution method.  The steps to solve 3 linear equations are as follow:
A.  Solve one equation for one of the variable
B.  Substitute the variable from A. into the 2 remaining equations.  You ended up having 2 equations with 2 variables.
C.  Solve one of the 2 equations from B. for one of the variable
D.  Substitute the variable from C. into the last equation from B.
E.  Solve the value for variable from D.
F.  Substitute value from E. into one of the equations from B.
G.  Solve the value of the remaining variable from B.
H.  Substitute the value from G. and E. into one of the equations from A. to find the value of the last variable

For the sake of using some concrete example, we will be solving the following equations:

equation #1:  4x + y + 6z = 7
equation #2:  3x + 3y + 2z = 17
equation #3:  -x - y + z = -9

We will solve for the variable z from equation #3.

equation #3':  z = -9 + x + y

Substitute #3' into equation 1 and 2.

equation #1':
4x + y + 6(-9 + x + y) = 7
4x + y - 54 + 6x + 6y = 7
10x + 7y = 61

equation #2':
3x + 3y + 2(-9 + x + y) = 17
3x + 3y - 18 + 2x + 2y = 17
5x + 5y = 35

Repeat substitution method for equation 1' and 2'

equation 1':  10x + 7y = 61
equation 2':  5x + 5y = 35

Solve 1' for x:
10x + 7y = 61
10x = 61 - 7y
x = 61/10 - 7/10y

NOTE OF CAUTION:  Do not change the fractions into decimal numbers.  The reason is that sometimes you will have a fraction such as x = 1/6 + 7/10y = 0.167 + .7y.  If you ended up rounding off 1/6 as 0.167, when you attempt to solve the rest of the equations for y and z, you will inevitably run into more rounding errors, hence not producing the correct answer.

Substitute x = 61/10 - 7/10y into equation 2':

equation 2':  5x + 5y = 35
5(61/10 - 7/10y) + 5y = 35
61/2 - 7/2y + 5y = 35
1.5y = 4.5
y = 3

Substitute y = 3 into equation #1' or #2':

equation 2':  5x + 5y = 35
5x + 5(3) = 35
5x + 15 = 35
5x = 20
x = 4

Substitute x = 4 and y = 3 into equation 1 or 2 or 3:

equation #3:  -x - y + z = -9
-4 -3 + z = -9
-7 + z = -9
z = -2

Therefore the solution to this system of linear equations is (4, 3, -2)

Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T:     201-444-8020   
E:  glenrock@mathnasium.com
www.mathnasium.com/glenrock