Algebra
Chapter 6 Answers
Systems of Equations and Inequalities
The triangulation process is used in many occupations. Cartographers use it while making maps; firefighters use it to pin point of forest fires. You can use information from the World Wide Web to plot triangulation points.
Part B, Systems of Linear Equations
1. Look at the Map of Paris Web site that shows the city's museums and monuments. Print the map and use it with an overhead transparency of 1 cm graph paper to complete the following problems. Make a coordinate axis on your graph paper.
a. If the Musée des Arts Decoratifs is the origin, what is the equation of the line containing Musée de l'Orangerie and Musée Auguste Rodin?
[y = x]b. If the Musée des Arts Decoratifs is the origin, what is the equation of the line containing the monuments Conciergerie and Place Vendome?
[y = -x]c. What is the relationship between the lines containing these points? Explain. Where do the lines intersect?
[The lines are perpendicular. Their slopes are negative reciprocals of each other. The lines intersect at #6 on the map representing the Place Vendome.]
Part C, Making Connections
2. Using your map from 1, plan a path from the Tour
Eiffel along a line with a slope of 2/3. What monument or museum
would you encounter? Explain your reasoning.
[From #17 on the map, move up 2 cm,
then right 3 centimeters. You would end up at #20, Place de la
Concorde.]
3. What would be the slope of the line containing the Musée
des Arts Decoratifs and the Cité des Science et de l'Industrie?
[The slope would be approximately 1.]
4. Using a pencil of a second color, draw a perpendicular
line to the line described in 2. If the origin is at #2,
what monument is on this perpendicular line? What are the coordinates
if your graph is in centimeters?
[Possible answers include: A perpendicular
line goes through the Galeries Nationales du Grand Palais. The
coordinates of this museum are approximately (-3, 1).]
The Ferris Wheel was the main attraction of the World's Fair in Paris in 1890. Go to the North Side: George Ferris Web site to read about this marvelous invention.
Part A, Solving by Substitution
1. Modern amusement parks and fairs have Ferris wheels too, but none equals the grandeur of the original Ferris Wheel built for the 1890 World's Fair. However, many new rides have been invented and are often combined with some old rides to create an amusement park for all ages. Open the pricing guide for the Santa Cruz Beach Boardwalk in California and plan a day of amusement.
a. Mary wants to take her family on two of the most famous rides at the boardwalk, the Ferris Wheel and the Giant Dipper roller coaster. Because the boardwalk is a busy place, Mary estimates that her family will have time for a total of only 15 rides. She has $35 to spend. Write a system of equations to represent Mary's situation. Let x = the number of Ferris Wheel rides and y = the number of Giant Dipper rides.
[x + y = 15 and $2.00 x + $3.00 y = $35.00]b. Solve this system of equations by substitution.
[x = 10 and y = 5]
Part B, Solving with Linear Combinations
2. , One of the employees at the ticket booth was extremely busy and lost track of the number of ticket strips and unlimited-ride wristbands were sold. José, the ticket manager, does know that the employee made a total of 500 sales, which totaled $10,525.
a. Write a system of equations to help José determine the number of each type of sale. Let x = the number ticket strips and y = the number of unlimited-ride wristbands.
[x + y = 500 and $24.95x + $18.95y = $10,525]b. Solve this system of equations using linear combination.
[ x = 175 and y = 325]c. Would you get the same results if you used the substitution method? Explain your thinking.
[Yes. Explanations will vary.]
Part C, Solving with Matrices
3. Use matrix multiplication to find the total amount of sales generated from each of two ticket booths if individual tickets cost $ .50 and wristbands cost $18.95.
|
Booth |
Individual Tickets |
Wristbands |
|
North |
2500 |
400 |
|
South |
1200 |
500 |
[At the north booth, ticket sales totaled $8830; at the south booth, ticket sales totaled 10,075.]
4. When do you think it would be easier to use matrix multiplication
rather than substitution or linear combinations to solve a system
of equations?
[Possible answers include: If a system
has more than two equations, it may be easier and faster to solve
this system using matrices and the graphing calculator.]
Part D, Making Connections
5. There are many ticket pricing options available at amusement parks. Look at the options at the Cedar Point Park in Sandusky, Ohio.
a. Use the data for admission prices. On one side of a file card, write a situation that could be solved using a system of equations. [Answers will vary.]
b. Solve your system of equations three ways: substitution, linear combination, and matrix multiplication. Write your solutions on a different file card.
[Answers will vary.]c. Trade cards with a classmate and solve his or her system of equations. Which solution method was easiest? Explain. [Answers will vary.]
Shadowgraphs and Shadowplay are artistic modes found throughout history. Another form of play dating back to the 19th century is tangrams.
Part A, Graphing Linear Inequalities
1. Browse through the site titled Tangram. Open the site New Tangram Puzzles and examine the puzzles. Refer to the Rectangle Tangram Puzzle and focus on the square tangram piece of this puzzle. Print a copy of the New Tangram Puzzles. Place the origin of a coordinate axes at the bottom right corner of the square piece of the Rectangle Tangram Puzzle.
a. Let the units along each axis be equal in length to the length of the sides of the square. What region is bounded by the inequality y < 1?
[The region is the entire rectangle.]b. Refer to the square piece of the Rectangle Tangram Puzzle and use the same origin and the same coordinate axes as in b. What region is bounded by the inequality x > 0?
[The right half of the rectangle.]
Part B, Graphing a System of Linear Inequalities
2. Refer to the square piece of the Rectangle Tangram Puzzle from 1. Again, place the origin of a coordinate axes at the bottom right corner of the square piece. Let one unit along each axis be equal in length to the length of the sides of the square.
a. What region is bounded by the axes and the inequalities y < 1 and x > -1?
[The sqaure piece]
b. What region is bounded by the axes and the inequalities y < 1 and x - y < 0?
[The small blue triangle]c. What inequalities describe the parallelogram tangram piece of the puzzle ?
[x - y >0, y >0, x - y <1, and y <1]
Part C, Making Connections
3. Working with a partner and using the New Tangram Puzzles, choose a point for the origin and draw a coordinate axes on a piece of graph paper.
a. Mark a convenient scale on each axis.
[Answers will vary depending on the origin.]b. Write a system of inequalities to describe each of the seven pieces of the Rectangle Tangram Puzzle.
[Answers will vary.]