1. Hawkes Learning Systems College Algebra
Section 8.1: Solving Systems by Substitution and Elimination

2. Objectives
Definition and classification of linear systems of equations.
Solving systems of equations using substitution.
Solving systems of equations using elimination.
Larger systems of equations.
Applications of systems of equations.

3. Linear Systems of Equations
Many problems of a mathematical nature are most naturally described by two or more equations in two or more variables. A collection of linear equations is called a linear system of equations, or simultaneous linear equations. The three graphs below pictorially identify the three possible varieties of systems of two linear equations in two variables.

4. Linear Systems of Equations
Varieties of Linear Systems
The chart below illustrates the only possible solutions to any linear system of equations. If a system of linear equations has
No Solutions
One Solution
Infinite Solutions
The system is called
Inconsistent
Consistent
Dependent

5. Solving Systems by Substitution
The solution method of substitution works by solving one equation and substituting the result into the remaining equations, a method that may require several repetitions. This method can be time consuming for large systems where there are many equations and variables, but for smaller systems, this method works well.

6. Solving Systems by Substitution
For example, we’ll demonstrate solving the system
by substitution.
We’ll show two (of the four possible) ways of solving this system of equations by substitution.
Possibility 1:
Possibility 2:
Choose an equation.
Solve for a variable.
Insert the solution into the other equation.
Solve for the other variable.
Solution is (2,3).

7. Solving Systems by Substitution
If we observe the graph of the two linear equations in the system we solved on the previous slide, we can see that the solution we found, (2,3), does indeed satisfy the system.

8. Example 1: Solving Systems by Substitution
Solve the system by substitution.
Choose an equation. (You could have chosen either!)
Solve for a variable. We choose y.
Insert the solution found for y into the other equation.
Solve for the other variable. In this case, x.
Thus, the solution is (1,1).

9. Example 2: Solving Systems by Substitution
Solve the system by substitution.
This is a true and trivial statement which implies that there are an infinite number of solutions. In other words, for any x-value, there exists a y-value that satisfies the system.

10. Solving Systems by Elimination
The method of elimination is based on the goal of eliminating one variable by adding two equations together (some texts call this the addition method). The elimination method works because of the following truth: If and then . Proof: We know that . Since , it must be true that . Similarly, so, . Thus, the above statement, “If and then ,” is true.

11. Solving Systems by Elimination
For example, we’ll demonstrate solving the following system by the method of elimination.
First, we’ll try adding the equations to see if any variables are eliminated.
Adding the equations as they are doesn’t help us, so let’s try manipulating one (or both) of the equations by multiplying both sides of that equation by a constant. If we multiply the first equation by ̶ 3 and add the equations again, the x variable will be eliminated.
Not helpful!
Continued on the next slide…

12. Solving Systems by Elimination
We found Now we can plug in y into either of the equations in the system. Thus, the solution to the system is .

13. Example 3: Solving Systems by Elimination
Solve the system by the method of elimination.
To eliminate a variable, you’ll need to multiply each equation by a unique constant. Let’s eliminate y. To do so, notice that we’ll have to multiply the first equation by 3 and the second equation by 2.
Thus, the solution is

14. Example 4: Solving Systems by Elimination
Solve the system by the method of elimination.
This equation is false, so the solution to this system is (there are no solutions).

15. Larger Systems of Equations
Algebraically, larger systems of equations can be dealt with in a similar manner to the systems we have been looking at. However, graphically, larger systems are very different. The table below highlights the differences between equations in three variables and equations in two variables.
An equation in
If a solution exists, it is called an
The graph contains
The graph is
2 variables
ordered pair
(x,y)
2 axes
2 dimensional
3 variables
ordered triple
(x,y,z)
3 axes
3 dimensional

16. Larger Systems of Equations
Below is a graphical representation of the ordered triple,

17. Example 5: Larger Systems of Equations
Solve the system
Select two equations and eliminate one variable. We’ll eliminate y from equations (I) and (II).
Continued on the next slide…

18. Example 5: Larger Systems of Equations(cont.)
Select two other equations and eliminate the same variable, y. We chose equations (I) and (III).
We have now found two linear equations in two variables. Now we will solve those equations by the elimination method.
Continued on the next slide…

19. Example 5: Larger Systems of Equations (cont.)
We know that z = –1. We’ll back substitute into one of the linear equations in two variables. We chose the second equation found.
Back substitute the values found into an original equation; we chose equation (I).

20. Example 6: Applications of Systems of Equations
Joey has $128 in his pocket in $10, $5, and $1 bills. There are 19 bills in all and he has four more $5 bills than $1 bills. How many bills of each kind does Joey have?
x = the number of $10 bills
y = the number of $5 bills
z = the number of $1 bills
Come up with a system of equations from the information given.
There are 19 bills.
The total value is $128.
There are four more $5 than $1 bills.
Continued on the next slide…

21. Example 6: Applications of Systems of Equations (cont.)
Now we can solve the system of equations.
Note: One of the equations in the system, y = z + 4 is already an equation in two variables, y and z.
Continued on the next slide…

22. Example 6: Applications of Systems of Equations (cont.)
Thus, Joey has nine $10 bills, seven $5 bills and three $1 bills.