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Math Review of Problem Solving with Systems of Equations

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Overview

Systems of two equations with two variables can also be used to solve problems.  In order to solve the problem, it can be translated to a system of equations.  Once the problem is understood, students can make a plan, find the answer, and check to make sure it is correct.

Understand the Problem

In order to understand the word problem, read it very carefully and note the questions asked, the data given.  Pay attention to any word cues that indicate mathematical relationships.  Suppose a basketball team played 180 games and they won 40 more games than they lost.  How many games did the team lose?  In that case, the team played 180 games total.  40 more games were won than lost.

Make a Plan

In making a plan, write a system of equations to fit the problem.  Let x be the number of games won and y be the number of games lost.  In the first equation, x +y = 180.  In the second equation, x –y =40.  Those equations can be combined to find the answer.

Find the Answer

This system of equations can be solved by the addition method, as x +y = 180 and x-y = 40.  Therefore 2x +(y-y) = 180 +40, or 2x= 220.  Solving for x, divide both sides by 2, so x =110.  Solving for y, 180 -110 = 70.

Check the Work

To see if both sentences are true, 110 +70 equals 180, and 110-70 equals 40. The same sort of process can be used to solve equations by choosing the substitution method.  Suppose that Matilde is 13 years older than Ana.  In 9 years, Matilde will be twice as old as Ana.  Let x be Matilde’s age and y be Ana’s age.  There are two equations in the plan, x =y +13, and x +9 =2(y +9).   Using substitution, x = y + 13, and x = 2y +9, so y +13 =2y +9, so y = 13-9, or 4.  Ana is 4, and Matilde is 17.  In 9 years Matilde will be 26, and Ana will be 4+9 or 13. Matilde is twice Ana’s age in 9 years, and the answers check.

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