Abstract: Grade/Subject: 9-12/Algebra

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Project AMP Dr. Antonio Quesada – Director, Project AMP

Grade/Subject: 9-12/Algebra
Time Frame: 5 or 6 – 40 minute periods
Strand: Algebra
Topic: Solving Systems of Linear Equations in Two Variables
Objectives: This lesson will help students learn the substitution and elimination methods of solving a linear equation by first using the graphing calculator to graph and construct tables of linear equations.
Materials: Graphing calculator, overhead display, handout (included)
Authors: Scott Waseman and Steve Donaldson


1. A linear system of equations in two variables will have no solution, one solution, or an infinite number of solutions; 2. Systems of linear equations model real-world phenomena.

Learning Objectives:

1. Student will be able to set up and solve systems of linear equations (algebra strand); 2. Student will be able to decide when a problem situation is best solved using a computer, calculator, paper and pencil, or mental arithmetic/estimation techniques (algebra).


Students will use a graphing calculator to solve linear systems of equations in two variables using graphing features and table features. Students will then learn the algebraic methods of substitution and elimination. Students will identify which method is most appropriate for a given system, and use these systems to solve practical applications. As a final assessment, students will set up, describe, and solve systems using all methods.


Informal observation and feedback (individual and group)

Demonstration of each method by student

Pencil and paper

Written analysis of methods

Learning Strategies:

This lesson lab is designed for algebra II students with a fundamental knowledge of using the graphing calculator. The first page of the included packet includes examples that can be shown to the students as appropriate; show them the graphing and table methods on the calculator (examples I & II) and then have them complete page 2 of the packet. Always allow them time to attempt the examples first individually or in small groups. The substitution method (example III) can be taught followed by page 3 of the packet, and elimination (example IV) followed by page 4 of the packet. The beginning of page 5 can be answered in groups and then summarized as a class. Pages 5 and 6 are intended to help the students determine which method is most appropriate and practice. After the students have a chance to work with each type, they can help come up with “weaknesses” for each technique. Page 7 provides practical application. Again, students should work in small groups to facilitate learning. Written assessment is also provided.

Classroom and Information Management:

This lesson is intended for rooms where students may view an overhead display and be able to move into groups (individual desks are ideal). Students should be at similar subject ability levels and instructor should be competent with demonstrating the examples. Allow ample time for students to attempt problems on their own.


Students are asked to find area and share methods of solution with classmates in discussion and in small groups.


Tables on the graphing calculator; graphs on the graphing calculator; evidence of knowledge on paper both mathematical and narrative.

Tools and Resources:

Graphing calculators, Overhead display, Worksheet with sample problems, Quiz

Do & How:

The graphing calculator will be used to help students understand relationships between two linear equations using a graph and table of values.

Page 1 Name

*A system of equations is solved by finding the (if any) that they share.

*A system of equations share either , , or points.
*A system of linear equations can be solved using one of four methods.

I. Graphing

II.. Table

III. Substitution

IV. Linear Combination


Examples of types I, II, III, and IV.
I. II. (use the table)


Page 2
Solve the following systems by graphing. Check your answers using a table.
1. 2.

3. 4.

5. 6.

Page 3
Solve the following systems using substitution.
7. 8.

9. 10.

11. 12.

Page 4
Solve the following systems using linear combination.
13. 14.

15. 16.

17. 18.

Page 5

Use the GRAPHING or TABLE method if…

Use the SUBSTITUTION method if…

Use the ELIMINATION (linear combination) method if…

For each of the following problems, describe briefly which method would be most appropriate and solve.
Sometimes you’ll have infinite solutions: Sometimes you’ll have no solution:

19. 20.

method: method

Some solutions are decimal: Some solutions are fractional:

21. 22.
method: method:

Page 6

Some equations are large:

23. 24.
method: method:

25. 26.

method: method:

Page 7
Use systems to solve the following problems. The choice of method is up to you. Use an alternate method to check your answers.

I. Some coins (dimes and nickels) are in a pile. The total

value of the pile is $1.35. The number of nickels is one

coin less than twice the number of dimes. Using a system

of equations, find the number of each type of coin.

II. The larger of two complementary angles is 12 more than

5 times the measure of the smaller. Find the measure of

the two angles.

III. A boat travels 224 miles upstream in 8 hours. The next day

It returns the same distance in 7 hours. Assuming the current

remained the same both ways, what was the rate of the boat

and the current?

IV. You are searching for two integers. The sum of twice the

first integer and three times the second integer is nine. At

the same time, the sum of three times the first integer and

twice the second integer is one. Find the two integers.

Linear Systems Test Name

  1. Solve by graphing. Check your proposed answer with a table. 1.

  1. Solve by substitution. 2.

  1. Solve by linear combination (elimination). 3.

Solve using the method of your choice.

4. Bill has $2.00 in quarters and dimes. The number 4.

of quarters is 4 less than twice the number of dimes.

Find the number of coins of each type.

5. The larger of two supplementary angles is 6 less than 5.

5 times the smaller. Find the measure of the two angles.

6. A plane flew 2,100 km with the jet stream in 2.5 hours. The 6.

return flight against the jet stream took 3.75 hours. Find

the speed of the jet stream and the airspeed of the plane.

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