MATH 117



CONTENT: Systems of Equations and their applications.

OBJECTIVES

Students should be able to:

1. Transform a system of equations to the form required to eliminate one of the variables.

2. Solve a system of equations using the method of elimination.

3. Translate real world application problems into systems of equations, solve them, and interpret the solution relative to the application.

4. Recognize the difference between the methods of substitution and elimination and determine which one is easier to use on a certain system of equations.

5. Recognize various kinds of real world problems that may be solved using a system of equations.

INTRODUCTION

Write down the equations of two straight lines or a system used in previous exmaples:

Eg. [pic]

Graph each equation separately by hand or using a graphing calculator.

Explain that each one is the equation of a straight line or parabola or whatever is in use, and each has infinitely many solutions as well as infinitely many non solutions.

Discuss with them the implications of using the conjunctions “and” and “or” in how we think about solutions.

When considered simultaneously (at the same time using “and”) the lines creates a “system of equations.” Have them guess a possible solution for the original problem. Then vary the problem to create systems with no solutions and infinitely many possible solutions.

Summarize by noting that systems can have one of the following solution sets (the solution points that make both equations true at the same time):

1. If the two lines are parallel, they have no solution in common.

2. If they intersect, they have one solution.

3. If they are the same line, they have infinitely many solutions.

4.

Challenge the students to guess a solution to motivate algebraic solving techniques

[pic]

Challenge students to determine where systems of equations fit in their daily life. Relate the following ideas after their responses.

1. Speed of a car, boats, airplanes, elevators, a bicycle, etc.

2. Mixture of acids, fuel, investments, sales, supply/demand, etc.

3.

LESSON:

Recall for them that in previous work they solved word problems using information like the following:

Jane has 35 coins, either nickels or dimes. She has $2.25 altogether in change. How many nickels does she have?

[pic]

Work through how we solved the problem for the variables using substitution.

Discuss the method of elimination as another algebraic method used to find the solution set to a system of equations.

1. Transform both equations so that they are in standard form (ax + by = c).

2. Make the coefficients of either the x variable or the y variable opposites

3. Review the additive property of zero using the commutative property of addition and the additive inverses (opposites) [pic]

4. Add the two equations, the variable with opposite coefficients will drop out and you will end up with one equation in one variable. This is very easy to solve for the variable that remains.

5. The value of the variable from step 3 is substituted in any of the original equations to end up with an equation in the other variable that can be solved in the same manner as in step 3.

6. Check that the solution set {all the (x, y) pairs} satisfies both equations.

7.

Activity

1. Do a step-by-step solution of a system of equations using the method of elimination. Involve students in every step by questioning them on the necessary steps.

2. Read out a word problem that is directly related to some idea that they are familiar with and solve it with their help. Interpret the answer relative to the original problem given.

3. Set up a few more systems of equations from other word problems as this is the hardest part of problem solving.

4.

Evaluation - Use a combination of the following:

5. Have students pair off and assign a word problem for them to talk their way through the steps of translating into symbolic form (equations), solve the system and interpret the answer.

6. Assign home work problems where they translate word problems. They then solve the system by the method of elimination and interpret their answers relative to the original problem.

7. Use an in class quiz with a simple word problem and a moderately difficult system.

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