Download Section 4.3 Solving Systems of Equations by Elimination (Addition)

Section 4.3
Solving Systems of Equations
by Elimination (Addition)
Objectives





Solve systems of linear equations by the elimination
method.
Use multiplication to eliminate a variable.
Use the elimination method twice to solve a system.
Use elimination to identify inconsistent systems and
dependent equations.
Determine the most efficient method to use to solve
a linear system.
Objective 1: Solve Systems of Linear
Equations by the Elimination Method

In the first step of the substitution method for solving
a system of equations, we solve one of the equations
for one of the variables.


At times, this can be difficult, especially if none of the
variables has a coefficient of 1 or −1. This is the case for
the system
Solving either equation for x or y involves working
with cumbersome fractions. Fortunately, we can
solve systems like this one using an easier algebraic
method called the elimination or the addition
method.
Objective 1: Solve Systems of Linear
Equations by the Elimination Method

The elimination method for solving a system is
based on the addition property of equality:
 When
equal quantities are added to both sides of an
equation, the results are equal.
In symbols, if A = B and C = D, then adding the left
sides and the right sides of these equations, we have
A + C = B + D. This procedure is called adding the
equations.
EXAMPLE 1

Solve the system:
Objective 1: Solve Systems of Linear
Equations by the Elimination Method

To solve a system of equations in x and y by the
elimination method, follow these steps.
1. Write both equations of the system in standard Ax + By = C
form.
2. If necessary, multiply one or both of the equations by a nonzero
number chosen to make the coefficients of x (or the coefficients of
y) opposites.
3. Add the equations to eliminate the terms involving x (or y).
4. Solve the equation resulting from step 3.
5. Find the value of the remaining variable by substituting the
solution found in step 4 into any equation containing both
variables. Or, repeat steps 2–4 to eliminate the other variable.
6. Check the proposed solution in each equation of the original
system. Write the solution as an ordered pair.
Objective 2: Use Multiplication to
Eliminate a Variable

In Example 1, the coefficients of the terms 5y in
the first equation and −5y in the second equation
were opposites.
 When
we added the equations, the variable y was
eliminated.

For many systems, however, we are not able to
immediately eliminate a variable by adding.
 In
such cases, we use the multiplication property of
equality to create coefficients of x or y that are
opposites.
EXAMPLE 2

Solve the system:
Objective 3: Use the Elimination Method
Twice to Solve a System

Sometimes it is easier to find the value of
the second variable of a solution by using
elimination a second time.
EXAMPLE 5

Solve the system:
Objective 4: Use Elimination to Identify
Inconsistent Systems and Dependent Equations
We have solved inconsistent systems and
systems of dependent equations by
substitution and by graphing.
 We can also solve these systems using the
elimination method.

EXAMPLE 6

Solve the system:
Objective 5: Determine the Most Efficient
Method to Use to Solve a Linear System

If no method is specified for solving a particular linear system, the
following guidelines can be helpful in determining whether to use
graphing, substitution, or elimination.

1. If you want to show trends and see the point that the two graphs have
in common, then use the graphing method. However, this method is not
exact and can be lengthy.
 2. If one of the equations is solved for one of the variables, or easily
solved for one of the variables, use the substitution method.
 3. If both equations are in standard Ax + By = C form, and no variable has
a coefficient of 1 or −1, use the elimination method.
 4. If the coefficient of one of the variables is 1 or −1, you have a choice.
You can write each equation in standard (Ax + By = C) form and use
elimination, or you can solve for the variable with coefficient 1 or −1 and
use substitution.
Objective 5: Determine the Most Efficient
Method to Use to Solve a Linear System

Here are some examples of suggested
approaches:
Objective 5: Determine the Most Efficient
Method to Use to Solve a Linear System

Each method that we use to solve systems of
equations has advantages and disadvantages.