Systems of Equations and Problem Solving – How to Set Up a System of Equations

There are two ways to solve a system of equations algebraically: the elimination method and the substitution method. Though the elimination method can be used at anytime, there are certain problems that lend them self to the use of the substitution method. Now that you have seen some examples on solving systems of equations with substitution, I can move on to some problem solving. The topic of this post is setting up and solving a system of two equations with the substitution method. The problem that follows is the classic part + part = whole relationship. The two parts are the cost of a slice of pizza and the cost of a soda. The whole is the total. Since there are two variables in the problem, there must be two equations. Read more to find out how to set up and solve a system of linear equations.

Example 1

At Renaldi’s Pizza, a soda and two slices of the pizza of the day costs $10.25. A soda and four slices of the pizza of the day costs $18.75. Find the cost of each item.

As you can see, the solution begins with the defining of variables: s for the price of one soda and p for the price of 1 slice of pizza. This is a fairly easy system of linear equations to set up because of the simple part + part = whole relationship. The two parts are the cost of the soda and the cost of the pizza. The whole is represented by the total cost. Thus, equations A and B are written based on the first two sentences of the problem.

Problem Solving - System of Equations - Pilarski

Problem Solving - System of Equations - Pilarski

Lines 3 and 4 are still equations A and B, but it is an equivalent system. The variable term in each equation was subtracted from each side to arrive at the new equations A and B. The equation in line 5 is from applying the substitution property to replace the variable s with the 10.25 – 2p from equation A in line 3. Line 6 is from by subtracting 4p and 10.25 from both sides of line 5. Finally, line 7 is the result of applying the division property of equality to line 6. The result: 1 slice of pizza (p) costs $4.25.

This problem is not completely solved. The cost of one soda is still unknown. To find the cost of the soda, the cost of a slice of pizza must be substituted into any of the equations A or B. It appears the original equation A is used in the diagram on line 8. Line 9 shows the substitution of 4.25 into an original equation. The equation in line 10 is possible by simplifying 2 times 4.25 and the result in line 11 is from subtracting 8.50 from both sides of 10.

The problem finishes up with a brief sentence explaining the results. I try to embed a video on all of my posts. The video is the same problem, but if you are an auditory learner, it might help you more to hear me saying the things I write about in my posts. Until next time.


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