A First Course in Linear Algebra, 2017a

10 Systems of Equations
Consider the following example.
Example 1.7: Effects of an Elementary Operation
Show that the system
has the same solution as the system
x + y = 7
2x − y = 8
x + y = 7
−3y = −6
Solution. Notice that the second system has been obtained by taking the second equation of the first system
and adding -2 times the first equation, as follows:
By simplifying, we obtain
2x − y +(−2)(x + y)=8 +(−2)(7)
−3y = −6
which is the second equation in the second system. Now, from here we can solve for y and see that y = 2.
Next, we substitute this value into the first equation as follows
x + y = x + 2 = 7
Hence x = 5andso(x,y)=(5,2) is a solution to the second system. We want to check if (5,2) is also a
solution to the first system. We check this by substituting (x,y)=(5,2) into the system and ensuring the
equations are true.
x + y =(5)+(2)=7
2x − y = 2(5) − (2)=8
Hence, (5,2) is also a solution to the first system.
♠
This example illustrates how an elementary operation applied to a system of two equations in two
variables does not affect the solution set. However, a linear system may involve many equations and many
variables and there is no reason to limit our study to small systems. For any size of system in any number
of variables, the solution set is still the collection of solutions to the equations. In every case, the above
operations of Definition 1.6 do not change the set of solutions to the system of linear equations.
In the following theorem, we use the notation E i to represent an expression, while b i denotes a constant.