Elimination Using Addition and Subtraction
Objective Introduce the elimination method of
solving systems of simultaneous equations.
The main idea in this lesson is that systems of equations may
be simplified and solved using the Addition and Multiplication
Properties of Equality to add and to subtract equations. The goal
is to obtain an equivalent equation in which only one variable is
on the left-hand side and only a number is on the right-hand
side.
The Elimination Method for Solving Systems of Equations
Let's start with some examples.
Example 1
Solve the system of linear equations.
x - y = 3
x + y = 7
Solution
For (x , y ) to be a solution to the system means that both
equations hold true for the values (x , y ). That is, both
equations are satisfied. Add the equations to get another valid
equation, using the Addition Property of Equality.
| x - y = 3 |
|
| (+) x + y = 7 |
Add the equations. |
| 2x + 0 = 10 |
|
| 2x = 10 |
|
This is a linear equation in one variable, namely x. Solve to
get x = 5. Then substitute x = 5 into one of the original
equations and solve for y.
| x - y |
= 3 |
|
| 5 - y |
= 3 |
Replace x with 5. |
| -y |
= -2 |
|
| y |
= 2 |
|
The only solution to the system of equations is (5, 2).
Example 2
Solve the system of equations.
2x + 5y = 7
2x - 2 y = 0
Solution
Use the Subtraction Property of Equality to subtract one of
the equations from the other to get another valid equation.
| 2x + 5y = 7 |
|
| (-) 2x - 2 y = 0 |
Subtract the equations. |
| 0 + 7y = 7 |
|
Solve 7y = 7 to get y = 1. Then substitute the value of y into
one of the equations and solve for x.
| 2x - 2(1) |
= 0 |
Substitute 1 for y. |
| 2x - 2 |
= 0 |
|
| 2x |
= 2 |
|
| x |
= 1 |
Divide each side by 2. |
The solution is (1, 1).
When the resulting equation involves only one variable, we say
that we have eliminated the other variable. In Example 1, y was
eliminated, and in Example 2, x was eliminated. Note to the
Teacher It is important for students to practice using this
method by working the exercises below. Then, introduce Examples 3
and 4, which are word problems.
Example 3
The sum of two numbers is 48, and their difference is 16. What
are the numbers?
Solution
Let x and y represent the two numbers and write the following
system of equations.
x + y = 48
x - y = 16
Adding the equations gives 2x = 64 or x = 32. To find the
value of y, substitute 32 for x in either equation to get y = 16.
The two numbers are 32 and 16.
|
