Download Pre-AP Algebra 2 Lesson 2-2

Pre-AP Algebra 2
Lesson 2-2 – Determining the number of solutions to a 2x2 system; Intro to 3x3 systems
Objectives: The students will be able to:
solve a 2x2 system of equations by elimination (review).
determine the number of solutions that a 2x2 system has, both algebraically and graphically.
solve a 3x3 system by elimination.
Materials: Hw #2-1 answers overhead; tally sheets; Do Now handout and answers overhead; note-taking templates;
pair work; homework #2-2
Time
5 min
10 min
20 min
25 min
Activity
Review Homework
Show the answers to #2-1 on the overhead. Students correct their answers. Pass around a tally sheet.
Homework Presentations
Review the top 2 or 3 problems.
Do Now
Pass out the How Many Solutions Are There? worksheet. Students solve and graph three 2x2 systems
of equations – one has 1 solution, one has no solutions, and one has infinitely many solutions. While
reviewing the answers on the overhead, help students make the connection between the number of
solutions and the number of intersection points.
Direct Instruction
Background Information:
A 2x2 System of Equations is a system of two equations with two variables that are solved together.
There are three possible outcomes with a 2x2 system:
Graph
Number of Solutions
Equation (c = any #)
Lines intersect at 1 point
1
x=c
Lines are parallel
None
0 = c (never true!)
Lines are the same
Infinitely many
0 = 0 (always true!)
A 3x3 System of Equations is a system of three equations with three variables that are solved together.
Concepts:
To solve a 3x3 system with linear combination:
1) Decide which variable is easiest to eliminate.
2) Eliminate that variable from one pair of equations.
3) Eliminate the same variable from a different pair of equations.
4) Write the resulting equations as a 2x2 system and solve it.
Notes:1) If the 2x2 has 0 or infinitely many solutions, the same is true about the 3x3 and
you can stop there. 2) See if you can use division to simplify the new equations.
5) Plug in the solutions to one of the original equations and solve for the third variable.
6) Check by plugging in the solution to the other two original equations.
Example:
Show students how to solve the system, alternating between the concept steps and the example steps.
20 min
3x
2x
y 2z
y 4z
5x
2y
z
8
(0, 2, 5) is the solution.
18
1
Pair Work
Hand out the Solving 3x3 Systems Practice sheet for students to work on.
Pre-AP Algebra 2
Lesson 2-2 – Bellringer
How Many Solutions Are There?
Directions:
1) Solve each system of equations algebraically. Two of them may seem to come out a little
strange.
Note: in each system, try to simplify the first equation by using division before you
do anything else.
2) Graph each system of equations on the coordinate planes below.
3) Compare your algebraic and graphical solutions. What conclusions can you make about
the relationship between the algebraic solution and the graph?
A)
2x 4 y 8
3x y 3
Conclusions:
B)
6x 3y
3
2x y
2
C)
2x 4 y
x 2y
8
4
Pre-AP Algebra 2
Lesson 2-2 – Pairwork
Solving 3 x 3 Systems of Equations
1) Solve the 2 x 2 system. Use division to simplify the equations before you begin.
4x 8y
20
77x 77 y 77
2) Multiple Choice. Which ordered triple represents the solution to the system of equations?
2x y 3z 11
x 2 y 4z
3
x 5y 2z
a. (-1, 1, -4)
b. (1, 3, -2)
c. (3, 8, 1)
d. (2, -3, 0)
18
3) Solve each system of equations. Label each equation and organize your work neatly.
2x 3y
z
3x
z
3x
2y
4y
5
5
2z
x 2y
z
2x 4 y
8
2
2z
4x 5y 7z
6
3
Pre-AP Algebra 2
Lesson 2-2 – Homework
3 x 3 Systems of Equations
Do all work on binder paper (stapled to the back). Write all answers on this page.
Part 1: Multiple Choice. What is the solution to the system of equations?
1)
2x y z 4
4x 2 y 2z 8
x 3y
z
5
a. (5, 4, -2)
2)
b. (0, -5, -1)
c. No Solution
d. Infinitely Many Solutions
c. No Solution
d. Infinitely Many Solutions
4x y z
3
2x 3y 4z 5
10x
y 5z
11
a. (-2, 3, -2)
b. (-1, 1, 0)
Part 2: Free Response. Solve each system of equations.
3)
2x y 3z 8
x 6y z 0
6x 3y 9z
24
Mega Bonus: +10 homework points!
4)
x 3y z
x 4 y 2z
3x
y
z
12
22
To solve a 4x4 system, use elimination to
reduce it to a 3x3 system, and then proceed
from there. If you want to get this problem
right, you need to stay super organized – but
everyone can do it if they try. You must show
all work to get credit.
16
w 2x
5)
y 2z
10
w 3x 2 y z 0
2w x y 4z 13
x 2 y z 12
x 2 y 3z 0
2x
y 4z
24
3w x 5y
z
7