Teaching Notes 3.19: Solving Systems of Equations Using
Multiplication with the Addition-or-Subtraction Method
Using multiplication with the addition-or-subtraction method is yet another way to solve systems
of equations. A trouble spot for students is deciding what number or numbers to multiply by.- Explain that using multiplication with the addition-or-subtraction method to solve systems
of equations involves two major steps: multiplying one or both of the equations by a nonzero
number to make the coefficients of one of the variables the opposite or the same and then
using the addition-or-subtraction method to solve the equations. - Present this example to your students: Solve 4x− 9 y=1and− 5 x+ 6 y=4. Ask them what
they would do to eliminate a term. Explain that there are two possibilities:- They could multiply the first equation by 2 and the second equation by 3 and add to elim-
inate they-terms. - They could multiply the first equation by 5 and the second equation by 4 and add to elim-
inate thex-terms.
Demonstrate how students may solve the system of equations using each method. The solu-
tion isx=−2andy=−1.
- They could multiply the first equation by 2 and the second equation by 3 and add to elim-
- Review the information and example on the worksheet with your students. Note that in the
example the method used eliminates thex-terms. Point out that the system could also be
solved by eliminating they-terms. The solution would be the same. (Note:The check is not
included with the example.) If necessary, review the addition-or-subtraction method with
your students.
EXTRA HELP:
After you have added or subtracted, one term should be eliminated. If no terms are eliminated,
check your work.ANSWER KEY:
(1)x=5;y= 3 (2)x=−1;y= 1 (3)x=−2;y=− 3 (4)x=3;y= 4 (5)x=5;y=− 2
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(Challenge)It is more efficient to eliminate thex-terms first because only the first equation would
be multiplied by 2. Choosing to eliminate they-terms first would require multiplying two equations
(the first by 5 and the second by 3). The solution isx=1andy=3.
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