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Copyright © 2011 Hawkes Learning Systems. All rights reserved.

Hawkes Learning SystemsCollege Algebra

Section 8.1: Solving Systems by Substitution and Elimination




Objectives

o Definition and classification of linear systems of equations.

o Solving systems of equations using substitution.o Solving systems of equations using elimination.o Larger systems of equations.o Applications of systems of equations.




Linear Systems of EquationsMany 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.




Linear Systems of Equations

Varieties of Linear SystemsThe 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

calledInconsistent Consistent Dependent




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.





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.

2 1

5

x y

x y

2 1x y 5x y Possibility 1: Possibility 2:

2 1y x

52 1x x

3 6x2x 2 2 1y

2 1x y

3

Choose an equation.

Solve for a variable.

Insert the solution into the other equation.

Solve for the other variable.

Solution is (2,3).

5x y

2 5 1yy

9 3y3y

5 3x

5x y

2





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.




Example: Solving Systems by Substitution

Solve the system by substitution.3 2 5

4 3

x y

x y

4 3x y

4 3y x

3 532 4xx

3 8 6 5x x 11 11x

1x 14 1 3y

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).




Example: Solving Systems by Substitution

Solve the system by substitution.2 4

3 6 12

x y

x y

2 4x y

2 4x y

3 2 4 6 12y y

6 12 6 12y y

0 0This 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.




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.

A B C D A C B D A C A C

A B A BC C C D A BC D

A B C DA C B D




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.

2

3 6 3

x y

x y

2y 6 3 3x y

x

4 5 1x y

3 3 3 6x y 3 6 3x y

9 9y

Continued on the next slide…

Not helpful!

3 6 3x y




Solving Systems by EliminationWe found

Now we can plug in y into either of the equations in the system.

Thus, the solution to the system is .

9 9y .1y

1 2x 1x

1, 1




Example: 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.

5 2 6

2 3 10

x y

x y

15 6 18x y 2 4 6 0x y

19 38x

5 2 6x y 2 3 10x y 32

2x 5 2 2 6y

2y Thus, the solution is . 2, 2




Example: Solving Systems by Elimination

Solve the system by the method of elimination.6

4 4 5

x y

x y

4 4 24x y + 4 4 5x y

0 29

6y 4 4 5x y

4

This equation is false, so the solution to this system is (there are no solutions).

x




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 inIf 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




Larger Systems of Equations

Below is a graphical representation of the ordered triple, . 2,1,3 -axisz

-axisy

-axisx

2,1,3

1 2

3 origin 0,0,0




Example: Larger Systems of Equations

Solve the system

Select two equations and eliminate one variable. We’ll eliminate y from equations (I) and (II).




Applications of Systems of Equationsa) How many ounces each of a 12% alcohol solution

and a 30% alcohol solution must be combined to obtain 60 ounces of an 18% solution?




Applications of Systems of Equationsb) A movie brought in $740 in ticket sales in one day.

Tickets during the day cost $5 and tickets at night cost $7. If 120 tickets were sold, how many were sold during the day?

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Documents

large systems

smaller systems

o larger systems of

o applications of systems

possible varieties of

linear system of equations

system of linear equations

simultaneous linear