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- Systems of linear equations
- Elimination by addition
Practice set a
Solve each system by addition.
Sample set b
Solve the following systems using the addition method.
Solve
Step 1: The equations are already in the proper form,
Step 2: If we multiply equation (2) by —3, the coefficients of
will be opposites and become 0 upon addition, thus eliminating
.
Step 3: Add the equations.
Step 4: Solve the equation
Step 5: Substitute
into either of the original equations. We will use equation 2.
We now have
and
Step 6: Substitute
and
into both the original equations for a check.
Step 7: The solution is
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Solve
Step 1: Rewrite the system in the proper form.
Step 2: Since the coefficients of
already have opposite signs, we will eliminate
.
Multiply equation (1) by 5, the coefficient of
in equation 2.
Multiply equation (2) by 2, the coefficient of
in equation 1.
Step 3: Add the equations.
Step 4: Solve the equation
Step 5: Substitute
into either of the original equations. We will use equation 1.
We now have
and
Step 6: Substitution will show that these values check.
Step 7: The solution is
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Practice set b
Solve each of the following systems using the addition method.
Addition and parallel or coincident lines
When the lines of a system are parallel or coincident, the method of elimination produces results identical to that of the method of elimination by substitution.
Addition and parallel lines
If computations eliminate all variables and produce a contradiction, the two lines of the system are parallel and the system is called inconsistent.
Addition and coincident lines
If computations eliminate all variables and produce an identity, the two lines of the system are coincident and the system is called dependent.
Sample set c
Solve
Step 1: The equations are in the proper form.
Step 2: We can eliminate
by multiplying equation (1) by –2.
Step 3: Add the equations.
This is false and is therefore a contradiction. The lines of this system are parallel. This system is inconsistent.
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Solve
Step 1: The equations are in the proper form.
Step 2: We can eliminate
by multiplying equation (1) by –3 and equation (2) by 4.
Step 3: Add the equations.
This is true and is an identity. The lines of this system are coincident.
This system is dependent.
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Practice set c
Solve each of the following systems using the addition method.
Exercises
For the following problems, solve the systems using elimination by addition.
Exercises for review
Questions & Answers
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Source:
OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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