Nicole Yingst MATH 395 Day 8: Cramer`s Rule a b c d = ad −bc 6 2 6 3

Nicole Yingst
MATH 395
Day 8: Cramer’s Rule
Day 8: Cramer’s Rule
Algebra 2: 9th Grade
Class period: 50 minutes
Goal: Students will be able to use Cramer’s Rule to solve a system of two linear equations.
Objectives:
1. Students will be able to evaluate the determinate of a 2x2 matrix created from a system of
two linear equations.
2. Students will be able to set up the two ratios used to find the solution to a system of two
linear equations through Cramer’s Rule.
3. Students will be able to find a solution to a system of two linear equations using Cramer’s
Rule and write the solution as an ordered pair.
4. Students will be able to examine and solve real life situations involving systems of linear
equations by using Cramer’s Rule.
Materials and Resources
Materials
1. Homework Copies
2. Group Handout Copies
3. Quiz Copies
Resources
1. Algebra 2 by Glencoe Mathematics (Teacher Wraparound Edition)
Lesson Procedure
1. Motivation: Bellwork (5 minutes)
A. Bellwork: Review Determinants through a short set of review questions. Bellwork will be
written on the board and students will write down and solve the following problems.
i.
Determinants:
a b
= ad − bc
c d
ii. Find the determinants of the following matrices. Show your work.
(1)
8 6
9 10
(2)
4 1
6 2
(3)
6 2
6 3
Nicole Yingst
MATH 395
Day 8: Cramer’s Rule
2. Review Bellwork Answers (5 minutes)
A. After students have finished the problems, reviewed the process and how to compute
the determinates, go over the answers to the bellwork.
B. Bellwork Key
i.
1.)
8 6
= 8(10) − 9(6) = 80 − 54 = 26
9 10
2.)
4 1
= 4(2) − 6(1) = 8 − 6 = 2
6 2
3.)
6 2
= 6(3) − 6(2) = 18 − 12 = 6
6 3
ii. This skill will be used a bit later in this lesson, so keep this in the back of your minds
as we shift to a different problem for a few more minutes.
iii. Opening Question: Find the solution to the following set of linear equations (3
minutes)
1)
5x + 7y = 13
2x − 5y = 13
2) Is this problem easy to complete with substitution and elimination?
3) Might there be an easier option?
3. Introduce Cramer’s Rule (7 minutes)
A. Cramer’s Rule uses matrices to solve systems of linear equations.
B.
ax + by = e
cx + dy = f
i.
b
d
a b
c d
,
y=
a
c
e
f
a b
c d
and
a b
c d
Question: Why is the last part necessary?
1) We can’t divide by zero, therefore the determinate cannon be equal to zero.
C. Let’s work through
i.
(x,y) where x =
e
f
x=
13 7
13 −5
5 7
2 −5
5x + 7y = 13
2x − 5y = 13 as a class.
=
13(−5) − 13(7) −65 − 91 −156
=
=
=4
5(−5) − 7(2) −25 − 14 −39
≠0
Nicole Yingst
ii.
y=
MATH 395
5 13
2 13
=
5 7
2 −5
Day 8: Cramer’s Rule
13(5) − 13(2) 65 − 29
39
=
=
= −1
5(−5) − 7(2) −25 − 14 −39
iii. Therefore, (x,y)=(4,-1)
D. Why does this work?
i.
Using Elimination solve for x
ax + by = e
cx + dy = f
adx + bdy = de
(−)bcx + bdy = bf
ii.
adx − bcx = de − bf
(ad − bc)x = de − bf
de − bf
x=
ad − bc
similarly
af − ce
y=
ad − bc
iii. The x = and y = relate to the determinate ratios that are a requirement of Cramer’s
Rule. Rewrite these ratios (or point to them if they are still on the board) to help
students see these connections.
E. Questions?
4. Class Example (5 minutes)
x − 4y = 1
2x + 3y = 13
x=
A.
y=
1 −4
13 3
1 −4
2 3
1 1
2 13
1 −4
2 3
(x, y) = (5,1)
=
=
1(3) − 13(−4) 3 − (−52) 55
=
=
=5
1(3) − 2(−4)
3 − (−8) 11
1(13) − 1(2) 13 − (2) 11
=
= =1
1(3) − 2(−4) 3 − (−8) 11
Nicole Yingst
MATH 395
Day 8: Cramer’s Rule
5. Group Work (20 minutes)
A. Groups will be made by the teacher. This will be done (for today) by sorting students by
grades high to low and numbering them off to get a reasonable mix of students in
each group.
B. Students will be asked to complete 3 real life examples of systems of two linear
equations that can be solved using Cramer’s Rule. The worksheet is attached, but
the problems are duplicated here:
i.
Stacy throws 2 bowls per hour and starts with 7 bowls. Her friend Nancy throws 5
bowls per hour and starts with 1 bowl. Using Cramer’s Rule and the following set
of linear equations, figure out how long it will take Stacy and Nancy to throw the
same number of bowls and how many bowls they will have each made.
−2x + y = 7
−5x + y = 1
ii. Steven ordered 8 shirts and 3 pairs of pants for $125.00. Matt ordered 6 shirts and 5
pairs of pants for $135.00. Using Cramer’s Rule and the following pair of linear
equations, figure out how much Steven and Matt paid for each shirt and pair of
pants.
8s + 3p = 125
6s + 5 p = 135
iii. Molly bought 3 points of apples and 2 pounds of pears for $6.81. Nate bought 2
pounds of apples and 4 pounds of pears for $9.02. Using Cramer’s Rule and the
following set of linear equations, figure out how much Molly and Nate paid per
pound for apples and pears.
3a + 2 p = 6.81
2a + 4 p = 9.02
6. Closure (5 minutes)
1. Review the answers to the group work.
2. Questions
i. Could we have solved these problems with substitution, elimination or by graphing?
(1) Yes, but they might have taken longer or been more difficult.
ii. Are there advantages and disadvantages to Cramer’s Rule?
(1) Yes, it takes a while, but sometimes it is a lot easier than dealing with the issues
caused by other methods.
7. Assessment (Quiz - Will be given during the next class after students have time to to ask
questions and practice this method through homework)
1. Students will be asked to complete 2 quiz problems to check for understanding of
Cramer’s Rule. Below are the two problems:
i.
2m + 7n = 4
m − 2n = −20
ii.
2r − s = 1
3r + 2s = 19
8. Pass out Homework Assignment. (Attached at the end)
Name:__________________ Date:__________________ Period:_________________
Day 8 Activity: Cramer’s Rule
Complete the following problems:
1. Stacy throws 2 bowls per hour and starts with 7 bowls. Her friend Nancy throws 5 bowls
per hour and starts with 1 bowl. Using Cramer’s Rule and the following set of linear
equations, figure out how long it will take Stacy and Nancy to throw the same
number of bowls and how many bowls they will have each made.
−2x + y = 7
−5x + y = 1
2. Steven ordered 8 shirts and 3 pairs of pants for $125.00. Matt ordered 6 shirts and 5
pairs of pants for $135.00. Using Cramer’s Rule and the following pair of linear
equations, figure out how much Steven and Matt paid for each shirt and pair of
pants.
8s + 3p = 125
6s + 5 p = 135
3. Molly bought 3 points of apples and 2 pounds of pears for $6.81. Nate bought 2 pounds
of apples and 4 pounds of pears for $9.02. Using Cramer’s Rule and the following
set of linear equations, figure out how much Molly and Nate paid per pound for
apples and pears.
3a + 2 p = 6.81
2a + 4 p = 9.02
Name:__________________ Date:__________________ Period:_________________
Day 8 Quiz: Cramer’s Rule
Use Cramer’s Rule to solve each system of equations.
1.
2m + 7n = 4
m − 2n = −20
2.
2r − s = 1
3r + 2s = 19
Name:__________________ Date:__________________ Period:_________________
Day 8 Homework: Cramer’s Rule
Use Cramer’s Rule to solve each system of equations.
1.
2m − 4n = −1
3n − 4m = −5
2.
4x + 3y = 6
8x − y = −9
3.
5x + 2y = 8
2x − 3y = 7
4.
9a − b = 1
3a + 2b = 12
5.
x + 5y = 14
−2x + 6y = 4
6.
3x + 4y = −15
2x − 7y = 19