614 Chapter 18 Mathematics in Engineering
Solutions of Simultaneous Linear Equations
As we discussed earlier, the formulation of many engineering problems leads to a system of
algebraic equations. As you will learn later in your math and engineering classes, there are a
number ways that we can use to solve a set of linear equations. In the section that follows, we
will discuss one of these methods that you can use to obtain solutions to a set of linear equations.Gauss Elimination Method We will begin our discussion by demonstrating the Gauss elimination
method using an example. Consider the following three linear equations with three unknowns:
x 1 ,x 2 , andx 3.(18.18a)
(18.18b)
(18.18c)
Step1: We begin by dividing the first equation, Equation (8.18a), by 2: the coefficient of the
x 1 term. This operation leads to(18.19)
Step2: We multiply Equation (18.19) by 3: the coefficient ofx 1 in Equation (18.18b).
(18.20)
We then subtract Equation (18.20) from Equation (18.18b). This step will eliminatex 1
from Equation (18.18b). This operation leads to(18.21)
Step 3: Similarly, to eliminatex 1 from Equation (18.18c), we multiply Equation (18.19) by 5:
the coefficient ofx 1 in Equation (18.18c).(18.22)
We then subtract the above equation from Equation (18.18c), which will eliminatex 1
from Equation (18.18c). This operation leads to(18.23)
a 5 x 1
5
2
x 2
5
2
x 3
65
2
b
7
2
x 2
1
2
x 3
31
2
5 x 1 x 2 3 x 3 17
5 x 1
5
2
x 2
5
2
x 3
65
2
a 3 x 1
3
2
x 2
3
2
x 3
39
2
b
1
2
x 2
5
2
x 3
25
2
3 x 1 2 x 2 4 x 3 32
3 x 1
3
2
x 2
3
2
x 3
39
2
x 1
1
2
x 2
1
2
x 3
13
2
5 x 1 x 2 3 x 3 17
3 x 1 2 x 2 4 x 3 32
2 x 1 x 2 x 3 13
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