Transcript 3.6Day1 - HonorsAlgebra2Meyer

3.6, Day #1
Systems of Linear Equations
with Three Variables
Do Now : “The Troll Toll”
1) Grab some slides. 2) Solve this brainteaser:
You are on your way to visit your Grandma, who lives
at the end of the valley. It's her birthday, and you
want to give her the cakes you've made.
Between your house and her house, you have to
cross 7 bridges, but there is a troll under every
bridge! Each troll, quite rightly, insists that you pay a
troll toll. Before you can cross their bridge, you have
to give them half of the cakes you are carrying, but
as they are kind trolls, they each give you back a
single cake.
How
many cakes do you have to leave home with
to make sure that you arrive at Grandma's with
exactly 2 cakes?
Do Now (Periods 2 and
10/11)
 G-g-g-g-graded
 Take
Do Now
some slides and the half-sheet that
says “3.5 Exit Card”. We did not have time
to do the Exit Card yesterday, so it will be
a graded “Do Now” instead. Please put
your stuff away and work on that. You will
have 5 minutes.
Clarification
 Standard
form of plane
ax+by+cz=d
a,b,c NOT all zero
This is why we could divide by d when we
did the problem with the intercepts and
the equation of the plane.
-Exit Card- Be sure to answer the questions
that are asked please. Write a function of
x and y means your answer should
include f(x,y)=…
Homework Concerns
 Were
there any?
That’s Space Ice Cream!
Weigh the Wangdoodles
http://www.mathplayground.com/wangdo
odles.html
Systems of Equations in
Three Variables
Solve this System
Solution
This is most easily done with substitution!
For instance, solve the last equation for z, z=3-2x+4y.
Now substitute that into the first equation.
2x+(3-2x+4y)=11
3+4y=11
y=2
Now we can plug that into the second equation to get x.
X+2(2)=7, x=3.
And similarly, we can now find z using the first equation.
2(3)+z=11, z=5
So the solution is (3,2,5) CHECK IT!
Let’s Do It!
ì3x + 2 y + 4z = 11
ï
í2x - y + 3z = 4
ï5x - 3y + 5z = -1
î
ì3x + 2 y + 4z = 11
ï
í2x - y + 3z = 4
ï5x - 3y + 5z = -1
î
Let’s Use Elimination!
Goal: System of 2 equations in 2 variables
Take any 2 of the equations, and eliminate
a variable
3x+2y+4z=11
(2x-y+3z=4)*2
Let’s eliminate y.
3x+2y+4z=11
4x-2y+6z=8
7x+10z=19
The first two equations gave us 7x+10z=19.
Now we can take the first and the third or
the second and the third and get another
equation with just “x”s and “z”s via
elimination. I will take the second and the
third.
(2x-y+3z=4)*-3
5x-3y+5z=-1
Now I have:
-6x+3y-9z=-12
5x-3y+5z=-1
-x-4z=-13
ì 7x +10z = 19
í
î-x - 4z = -13
Finishing it Up!
ì 7x +10z = 19
í
î-x - 4z = -13
Solve to see x=-3
and z=4. Plug these
back into any
equation to get
y=2. The solution is
then (-3,2,4).
CHECK IT IN ALL 3
EQUATIONS!
You Try!
ì x + 5y - z = 16
ï
í3x - 3y + 2z = 12
ï2x + 4 y + z = 20
î
What Do You Notice?
ìx + y + z = 2
ï
í3x + 3y + 3z = 14
ïx - 2 y + z = 4
î
ìx + y + z = 2
ï
íx + y - z = 2
ï2x + 2 y + z = 4
î
ìx + y + z = 2
ï
í3x + 3y + 3z = 14
ïx - 2 y + z = 4
î
 Multiply
the top equation by -3 and
combine with the middle equation. We
end up with 0=8. NONSENSE! No solution!
ìx + y + z = 2
ï
íx + y - z = 2
ï2x + 2 y + z = 4
î
Combine the first and the second and get
2x+2y=4
Combine the second and the third and get
3x+3y=6
These are both just multiples of x+y=2. If we
combined them, we would get 0=0.
Infinitely many solutions!
Application

A public swimming pool has the following rates:
ages under 5 are free, ages 5-16 are $3, and ages
16 and up are $4. The pool also has a policy that
every child under age 5 must be accompanied by
an adult. The families in your neighborhood
decide to go to the pool as part of a summer
party. There are 22 people in your group and an
equal number of children under age 5 as people
16 years old and older. The total admission cost
was $54. How many of each group went?

X = ages under 5
Y = ages 5 -16
Z = ages 16 and up


So there are 22 people in the group.
This means:
x+y+z=22
The total cost was $54 with the prices for each age group given
to give:
3y+4z=54
Finally, there are an equal number of children under 5 as people
16 and over so that:
x=z
ì x + y + z = 22
ï
í3y + 4z = 54
ïx = z
î
x=6, y=10, z=6
So 6 people
under 5, 10
people 5-16
and 6 people
16 and up
You Try!
 Courtney
has a total of 256 points
on three Algebra tests. His score
on the first test exceeds his score
on the second by 6 points. His
total score before taking the third
test was 164 points. What were
Courtney’s test scores on the
three tests?
Answer
 First=85,
second=79, third=92
Chapter 3 Survey
Other Games
 http://www.mathplayground.com/algebr
aic_reasoning.html
 http://www.mathplayground.com/algebr
a_puzzle.html