Download EQUATIONS IN THREE VARIABLES Find the solution of each

EQUATIONS IN THREE VARIABLES
Find the solution of each system.
1.
xy z6
2 x  y  3z  9
 x  2 y  2z  9
2.
xy z4
x  2y  z  1
2 x  y  2z  1
3.
2 x  y  3z  1
2 x  y  z  9
x  2y  4z  17
4.
2 x  y  z  10
4 x  2y  3z  10
x  3 y  2z  8
5.
2x  3y  z  5
x  3 y  8z  22
3 x  y  2z  12
6.
4x  y  z  4
2x  y  z  1
6 x  3y  2z  3
7. The sum of three numbers is 105. The third number is 11 less than 10 times the second. Twice
the first is 7 more than 3 times the second. Find the numbers.
8. The sum of three numbers is 5. The first number minus the second plus the third is 1. The first
minus the third is 3 more than the second. Find the numbers.
9. In Triangle ABC, the measure of angle B is twice the measure of angle A. The measure of angle C
is 80° more than that of angle A. Find the angle measures.
10. Linda has a total of 225 on three tests. The sum of the scores on the first and second test
exceeds her third score by 61. Her first score exceeds her second by 6. Find the three scores.
11. Jim has 28 coins made up of nickels, dimes and quarters. He has four more dimes than nickels
and quarters combined. How many of each kind of coin has he if the total value is $3.20?
SYSTEMS OF EQUATIONS USING MATRICES
ON THE GRAPHING CALCULATOR
Example:
x  2y  3z  11
2 x  y  2z  9
4 x  3 y  z  16
Equations must be written in standard form!!!
Press 2nd MATRIX and use the right or left arrow to get to EDIT. Press 1 for matrix A. Adjust the
dimensions of matrix A to 3 X 3 using ENTER or the arrow keys to move to the locations needing
adjustment. Move down through the list of elements entering the COEFFICIENTS of the variables from
each equation.
 1 2 3 
 A  2 1 2
 4 3 1 
Press 2nd MATRIX and use the right or left arrow to get to EDIT. Press 2 for matrix B. Adjust the
dimensions on matrix B to 3 X 1. Move down through the list of elements entering the CONSTANTS of the
equations.
11
B    9 
16 
Press 2nd QUIT to return to the home screen. Press CLEAR. To obtain the solution, press 2nd MATRIX, 1
to matrix A. Press the x 1 key to indicate the inverse of matrix A. Press 2nd MATRIX, 2 to obtain matrix
B. Press ENTER. The matrix displayed vertically represents the solutions for x, y and z.
 A  B   the solution
1
For example
x  2 
y    3 
   
 z   1
This works for a system of 2 equations in 2 variables too! Try it!
Magnet Advanced Algebra
Matrices Activity Sheet
Name ________________________________
Date _______________
Block ________
1. The Yummy Pizza Company has two locations: one at the Lynnhaven Mall and one
at the Pembroke Mall. On July 4, the Lynnhaven Mall store sold 231 small pizzas, 452
medium pizzas, and 186 large pizzas. Also on July 4, the Pembroke Mall store sold
242 small, 258 pizzas, and 241 large pizzas.
a) Represent the above information in a July 4 sales matrix, J. (Hint: pizza size by location)
 231 242 
J   452 258 
186 241 
July 4th matrix
b) Suppose that on July 5, the Lynnhaven Mall store sold 285 small pizzas, 269 medium pizzas, and 530
large pizzas. On the same day the Pembroke Mall store sold 200 small ones,
187 medium ones, and 109 large ones. Represent this information in the same kind of
 285 200 
matrix called K.
K   269 187 


 530 109 
July 5th matrix
c) Find J + K and describe in a complete sentence what this matrix tells you.
 516 442 
J  K   721 445 
 716 350 
July 4th and July 5th sales combined
d) If the manager of the Lynnhaven Mall store expects next year’s pizza sales rise by 6.5%, about how
many large pizzas does the manager expect to sell at both stores next July 4? If you were the manager,
would you round your calculations up or down? What scalar multiple did you use to assist you in planning
next July 4th’s sale?
1.065 186 241  199 257 (rounded up)
2. Tickets to the basketball tournament cost $2.00 for students, $5.00 for adults,
and $4.00 for senior citizens. At the Friday night game, there were 921 students, 174
adults, and 43 senior citizens. At the Saturday night game, there were 1023 students,
153 adults, and 25 senior citizens. Put this information in matrix form. Use matrix
multiplication to find the income from the ticket sales for the Friday and Saturday
night games.
 921 1023 
2
3.
5 4 174
 43
153    2884 2911
25 
can also make a 3 x 1 and a 2 x 3…..
At Burger Heaven, 2 cheeseburgers and 3 orders of fries cost $3.65. A
cheeseburger and 2 milkshakes cost $2.47. A cheeseburger, 2 orders of fries,
and a milkshake cost $3.01. Using matrices, determine the cost of each item.
x = cheeseburgers
y = fries
z = milkshakes
1
 2 3 0   3.65   .79
 1 0 2   2.47    .69 

 
  
 1 2 1   3.01  .84
2x + 3y = 3.65
x + 2z = 2.47
x + 2y + z = 3.01
Therefore, cheeseburgers cost $.79
fries cost $.69 and milkshakes cost $.84
4. Matrix S gives the number of three types of cars sold in October by Beach Ford and RK Chevrolet, and
matrix P gives the profit for each size of car sold.
Dealers
RK
BF
18 17 
 S
Mid-size  44
13


Compact 10
33 

Full-size
Compact
Profit
$400
Mid-size
Full-size
$650 $900  P
a)
Are there any problems with the way the matrices have been set up?
Yes, the type of car needs to be in the same order for both matrices
b) Which matrix is defined, SP or PS? Why?
PS because the inner dimensions match
c) Find either SP or PS and interpret the result. (What does the result mean when you multiply these
two matrices?)
18 17 
 900 650 400   44 13    48800 36950
10 33 
5. Ms. Coldstone owns three ice-cream shops and wants to know how much she made on
ice-cream cone sales for one day. The number of small, medium, and large cones sold at
each location and the profit for each size are contained in the tables.
Va Beach
Norfolk
Chesapeake
Small
74
32
120
Medium
25
38
52
Large
37
16
34
Small
Medium
Large
Profit
$0.90
$1.25
$2.15
a) Write a quantity matrix that gives the number of each sold at each location and a profit matrix that
gives the profit for each size. What must be the same for each matrix?
Same info written in matrices instead of tables……inner dimensions must match so that the
columns of the first matrix represent size and the rows of the 2nd matrix also represent size
b) Find the profit from ice-cream cones for each location. What did you do to get the answer?
 177.40 
 110.70 


 246.10 
c) What are the dimensions of the answer matrix? What do the entries in the answer matrix tell you?
Convert your matrix into a table with row and column headings so that Ms. Coldstone can understand the
information. 3 x 1, profit at each location…..