7.3
Solving Linear Systems by
Linear Combinations
What you should learn
GOAL 1 Use linear
combinations to solve a
system of linear equations.
GOAL 2 Model a real-life
problem using a system of
linear equations, such as
the mixture problem in
Example 4.
GOAL 1
Sometimes when you want to solve a linear system, it is not easy to isolate one of
the variables. In that case, you can solve the system by linear combinations. A
linear combination of two equations is an equation obtained by adding one of
the equations (or a multiple of one of the equations) to the other equation.
S O LV I N G A L I N E A R S Y S T E M B Y L I N E A R C O M B I N AT I O N S
Why you should learn it
FE
䉲 To solve real-life problems
such as finding the speed
of the current in a river
in Ex. 48.
AL LI
RE
USING LINEAR COMBINATIONS
STEP 1
Arrange the equations with like terms in columns.
STEP 2
Multiply one or both of the equations by a number to obtain
coefficients that are opposites for one of the variables.
STEP 3
Add the equations from Step 2. Combining like terms will
eliminate one variable. Solve for the remaining variable.
STEP 4
Substitute the value obtained in Step 3 into either of the
original equations and solve for the other variable.
STEP 5
Check the solution in each of the original equations.
EXAMPLE 1
Using Addition
Solve the linear system.
4x + 3y = 16
2x º 3y = 8
Equation 1
Equation 2
SOLUTION
1
The equations are already arranged.
2
The coefficients for y are already opposites.
3
Add the equations to get an equation in one variable.
4x + 3y = 16
2x º 3y = 8
Write Equation 1.
Write Equation 2.
6x
Add equations.
= 24
x=4
4
Substitute 4 for x in the first equation and solve for y.
4(4) + 3y = 16
y=0
5
䉴
Solve for x.
Substitute 4 for x.
Solve for y.
Check by substituting 4 for x and 0 for y in each of the original equations.
The solution is (4, 0).
7.3 Solving Linear Systems by Linear Combinations
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