Project AMP
Dr. Antonio Quesada – Director, Project AMP
Abstract:
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
Concept:
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).
Project/Task:
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.
Assessment:
Informal observation and feedback (individual and group)
Demonstration of each method by student
Pencil and paper
Written analysis of methods
Project AMP
Dr. Antonio Quesada – Director, Project AMP
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.
Sharing:
Students are asked to find area and share methods of solution with classmates in
discussion and in small groups.
Results:
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.
Dr. Antonio Quesada – Director, Project AMP
Project AMP
Page 1
Name
*A system of equations is solved by finding the
*A system of
equations share either
(if any) that they share.
,
, or
points.
*A system of linear equations can be solved using one of four methods.
I.
II..
III.
TYPE
Graphing
DEF’N
WEAKNESSES
Table
Substitution
IV. Linear Combination
(Elimination)
Examples of types I, II, III, and IV.
I.
x  y  0

2 x  y  3
x  y  1
III. 
x  2 y  7
x  y  0
II. 
2 x  y  3
3x  4 y  20
IV. 
4 x  5 y  10
(use the table)
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Page 2
Solve the following systems by graphing. Check your answers using a table.
y  2x  3
1. 
 y  5 x  2
 y  3x  3
2. 
 y  3x  1
2 y  4 x  6
3. 
4 y  8 x  12
 y  21 x  4
4. 
2 y  x  6
x  y  1
5. 
 x  y  1
x  y  0
6. 
x  y  1
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Page 3
Solve the following systems using substitution.
y  x  1
7. 
x  1  y
y  4x  1
8. 
2 x  y  37
3x  y  5
9. 
x  y  9
 5m  9n  21
10. 
2m  2n  14
4a  7b  34
11. 
2b  8a  4
11x  4 y  14
12. 
 2 x  y  16
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Page 4
Solve the following systems using linear combination.
2 x  7 y  40
13. 
2 x  3 y  20
 3 x  4 y  7
14. 
2 x  y  1
8 x  1
15. 
  3 x  y  4
 x  y  6
16. 
7 x  3 y  18
11x  2 y  27
17. 
21x  3 y  9
3x  8 y  10
18. 
 21x  56 y  84
Project AMP
Dr. Antonio Quesada – Director, Project AMP
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:
 y  17  x
19. 
3x  3 y  51
Sometimes you’ll have no solution:
3x  y  4
20. 
3x  y  1
method:
method
Some solutions are decimal:
3x  2 y  47.5
21. 
.
5x  4 y  2713
Some solutions are fractional:
4 x  y  7
22. 
1
x  2 y  2
method:
method:
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Page 6
 y  3x  2
23. 
x  y  2
Some equations are large:
100 x  100 y  500
24. 
200 x  100 y  100
method:
method:
3a  2b  14
25. 
a  3b  10
4m  2n  2
26. 
5m  n  8
method:
method:
Project AMP
Dr. Antonio Quesada – Director, Project AMP
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.
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Linear Systems Test
Name
1. Solve by graphing. Check your proposed answer with a table.
xy3
 3 x  y  5
1.
2. Solve by substitution.
4 x  y  13
x  4y  1
2.
3. Solve by linear combination (elimination).
2 x  3 y  25
3 x  4 y  5
3.
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Solve using the method of your choice.
4. Bill has $2.00 in quarters and dimes. The number
of quarters is 4 less than twice the number of dimes.
Find the number of coins of each type.
4.
5. The larger of two supplementary angles is 6 less than
5 times the smaller. Find the measure of the two angles.
5.
6. A plane flew 2,100 km with the jet stream in 2.5 hours. The
return flight against the jet stream took 3.75 hours. Find
the speed of the jet stream and the airspeed of the plane.
6.