Systems of Linear Equations Algebra I by jennyyingdi

VIEWS: 12 PAGES: 21

									Algebra I – Turner
Chapter 7




Lesson Goal: Solve a system of two linear equations by graphing.


Definition       System of Equations: A system of equations consists of two or more
                 equations in two or more variables.

Definition       Solution of a System of Equations: A solution of a system of equations in
                 two variables is an ordered pair that makes every equation in the system true.



1.   Consider the system: 5x  y  8 and 3x  2 y  2 .

     a.     Is (6, 3) a solution?                         b.   Is (1,  3) a solution?




     c.     Is ( 2, 2) a solution?




2.   Consider this system of equations: x  y  7 and x  y  3 . Can you find some solutions
     by trial and error?


                                                                                     y
                                                                                8
                                                                                7
                                                                                6
                                                                                5
                                                                                4
3.   Solve graphically: x  y  7 and x  y  3 .                               3
                                                                                2
                                                                                1
                                                          -8 -7 -6 -5 -4 -3 -2 -10       1 2 3 4 5 6 7 8
                                                                                                           x
                                                                                -1
                                                                                -2
                                                                                -3
                                                                                -4
                                                                                -5
                                                                                -6
                                                                                -7
                                                                                -8
Algebra I – Turner                                                                            Classwork – 7

4.   Could a system of linear equations ever have more than one solution? Could it ever have no
     solutions?




                                                                                        y
5.   Solve graphically: 2 x  3 y  6 and 4 x  3 y  12 .                         8
                                                                                   7
                                                                                   6
                                                                                   5
                                                                                   4
                                                                                   3
                                                                                   2
                                                                                   1
                                                             -8 -7 -6 -5 -4 -3 -2 -10       1 2 3 4 5 6 7 8
                                                                                                              x
                                                                                   -1
                                                                                   -2
                                                                                   -3
                                                                                   -4
                                                                                   -5
                                                                                   -6
                                                                                   -7
                                                                                   -8



                                                                                        y
                                             1                                     8
6.   Solve graphically: 3 y  x  9 and y   x  1 .                              7
                                             3
                                                                                   6
                                                                                   5
                                                                                   4
                                                                                   3
                                                                                   2
                                                                                   1
                                                             -8 -7 -6 -5 -4 -3 -2 -10       1 2 3 4 5 6 7 8
                                                                                                              x
                                                                                   -1
                                                                                  -2
                                                                                   -3
                                                                                  -4
                                                                                  -5
                                                                                  -6
                                                                                  -7
                                                                                  -8




                                                 2
Algebra I – Turner                                                                             Classwork – 7

                                 2                                                      y
7.   Solve graphically: y          x  1 and x  4 .                               8
                                  3                                                 7
                                                                                    6
                                                                                    5
                                                                                    4
                                                                                    3
                                                                                    2
                                                                                    1
                                                              -8 -7 -6 -5 -4 -3 -2 -10       1 2 3 4 5 6 7 8
                                                                                                               x
                                                                                    -1
                                                                                   -2
                                                                                    -3
                                                                                   -4
                                                                                   -5
                                                                                   -6
                                                                                   -7
                                                                                   -8


How many solutions would each system have?

8.    y  x  4           and        y  x  9


9.    y  2x  4           and        y  3x  6


10. y  6 x  2            and        y  6x  2




Lesson Goal: Solve a system of two linear equations by substitution.


Process          Solving a System of Equations by Substitution
                 1.   Solve one equation for one variable.
                 2.   Substitute this expression into the other equation.
                 3.   Solve the resulting equation.
                 4.   Substitute this value into the expression obtained from step #1.
                 5.   Write the solution set.




                                                   3
Algebra I – Turner                                              Classwork – 7

11. Solve by substitution: 2 x  y  9 and 7 x  3 y  1 .




12. Solve by substitution: 4 x  y  15 and x  y  10 .




13. Solve by substitution: 2 x  4 y  7 and  2 y  x  1 .




                                                4
Algebra I – Turner                                                                 Classwork – 7

14. Solve by substitution: 5x  3 y  15 and x  3 y  7 .




Lesson Goal: Solve a word problem using a system of linear equations.

15. A quilt maker sews both large and small quilts. A large quilt requires 8 yards of fabric
    while the small quilt requires 3 yards. How many of each size quilt did she make if she
    used a total of 90 yards of fabric to make 15 quilts?




16. A store is selling all shoes for $12 a pair and all slippers for $8 a pair. If Linda buys a
    total of 7 items and spends a total of $76, how many pairs of shoes and slippers did she
    buy?




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Algebra I – Turner                                                                  Classwork – 7

17. A toy maker produces wooden trains and wooden airplanes. Each train requires 3 ounces
    of paint and each airplane requires 5 ounces of paint. The toy maker has a gallon can of
    paint (64 ounces). If he wants to paint 14 toys, how many of each can he paint?




18. A girl has 15 coins, all nickels and dimes, with a total value of $1.20. Find the number of
    each kind of coin.




Lesson Goal: Solve a system of two linear equations by addition.


Review:
Theorem          Addition Property for Equality: If a  b , then a  c  b  c .
Restatement:
Theorem          Addition Property for Equality: If a  b and c  d , then a  c  b  d .

Now consider it written in this form:
  ab
() c  d
ac  bd

This is the form of the property that we are going to use in our solutions today.




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Algebra I – Turner                                                               Classwork – 7

19. Solve by addition: x  y  7 and x  y  1. Then check.




20. Solve by addition:  2a  b  15 and 2a  3b  5 .




21. Solve by addition: 5x  3 y  32 and 5x  7 y  8 .




22. A plumber charges a flat fee to work on a job plus an hourly rate. If he charged $150 for a
    3 hour job and $300 for an 8 hour job, what was the flat fee and the hourly rate?




                                               7
Algebra I – Turner                                                           Classwork – 7

Lesson Goal: Solve a system of two linear equations by addition and multiplication.
23. Solve by linear combinations: 3m  4n  2 and 5m  n  12 .




24. Solve by linear combinations: 4r  14  5t and 3r  6t  9 .




25. Solve by linear combinations: 7 x  14  2 y and 28  4 y  14 x .




26. The winning average on a Junior High baseball team was 0.381. If the team had won 3
    more games, then its winning average would have been 0.524 . Find the number of games
    the team won and the number of games it played.




                                               8
Algebra I – Turner                                                                       Classwork – 7

27. The difference between three times one number and a smaller number is 23. The sum of
    the smaller and twice the larger number is 27. Name the numbers.




Lesson Goal: Solve interest problems using a system of linear equations.

Using the Language of Investments

Review:
Formula          Interest: Interest  Principal  Rate  Time



Investment problems use a variety of words to describe interest, principal, and rate. Here are
some of the words you should associate with each:

       Principal always refers to the starting amount of money that is invested, deposited, or
        borrowed.

       Rate is sometimes referred to as interest, but it is easily identified as the rate because it is
        always expressed as a percentage.

       Interest is the amount of money that is earned by the investment. It can be referred to as
        the yield, dividend, growth, increase, return, earnings, or income.


28. Mr. Kirchner invested $8000, part earning 8% interest and the rest earning 10% interest
    per year. How much did he invest at each rate if his annual yield was $760?




                                                   9
Algebra I – Turner                                                               Classwork – 7

29. Mrs. Mooney wants to earn at least $288 in annual dividends. She plans to invest a certain
    sum of money at 12% and 3 times as much at 8%. What is the least amount she can
    invest at 12% to accomplish this goal?




30. Mrs. Miller deposited $1400, part at 9% and the rest at 12% annual interest. If both
    deposits earn the same annual income, how much has been deposited at each rate?




31. Miss Dilmer took out two loans to pay for a car. One loan required her to pay 10% annual
    interest while the other required her to pay 15% annual interest. Miss Dilmer borrowed
    $1400 more at the lower rate than she did at the higher rate. If the annual interest on the
    loan at 10% was $25 less than the interest on the loan at 15%, how much did she borrow
    at each rate?




                                              10
Algebra I – Turner                                                                      Classwork – 7

Lesson Goal: Solve word problems involving wind and water currents using a system
of linear equations.

Wind and Water Currents


Application:     Water Currents: When traveling upstream, the actual rate a boat is traveling
                 equals the rate of the boat in still water minus the rate of the current. When
                 traveling downstream, the actual rate a boat is traveling equals the rate of the boat
                 in still water plus the rate of the current.

Example: If Bob can row his boat at 8 km/hr in still water and he is rowing upstream against a
current traveling at 3 km/h, then his boat is actually traveling up the river at 8  3  5 km/h.

Example: If Bob can row his boat at 8 km/hr in still water and he is rowing downstream with a
current traveling at 3 km/h, then his boat is actually traveling up the river at 8  3  11 km/h.


Application:     Wind: When traveling against the wind (called flying with a headwind), the
                 actual rate an airplane is traveling (called the groundspeed) equals the rate of the
                 airplane in still air (called the airspeed) minus the rate of the wind. When
                 traveling with the wind (called flying with a tailwind), the actual rate an airplane
                 is traveling equals the rate of the airplane in still air plus the rate of the wind.

Example: If a plane travels at a still air rate (or airspeed) of 300 km/h against a 5 km/h head
wind, then the actual rate the plane is traveling (or groundspeed) is 300  5  295 km/h.

Example: If a plane travels at a still air rate (or airspeed) of 300 km/h with a 5 km/h tail wind,
then the actual rate the plane is traveling (or groundspeed) is 300  5  305 km/h.

32. An airplane travels 1800 miles in 3 hours flying with the wind. On the return trip, flying
    against the wind, it takes 4 hours. Find the rate of the wind and the rate of the plane in still
    air?




                                                  11
Algebra I – Turner                                                                 Classwork – 7

33. A barge travels 24 miles up the Ohio River in 3 hours. The return trip takes 2/3 that
    amount of time. Find the rate of the barge in still water and the rate of the current.




34. Len is planning a five-hour flight up to the mountains and back. He knows that he can fly
    in still air at 100 mi/h and finds that he has a 10 mph tail wind as he heads towards the
    mountains. How much time can he spend flying towards the mountains before he must turn
    back?




35. A rowing team rows downstream for half an hour, turns around, and rows back to their
    starting point in an hour and a quarter. If the current flows at 3 miles per hour, how fast
    does the team row, and how far did they travel downstream?




                                               12
Algebra I – Turner                                                                  Classwork – 7

Lesson Goal: Solve distance-rate-time word problems using a system of linear
equations.

36. Two ships are sailing toward each other from an initial distance of 120 nautical miles
    apart. The rate of one ship is 4 knots greater than the rate of the other ship. If they meet in
    3 hours, find the rate of each ship.




37. A motorist traveling 55 miles per hour is being pursued by a highway patrol car traveling
    65 miles per hour. If the patrol car is 4 miles behind the motorist, how long will it take
    the patrol car to overtake the motorist?




38. Sally started home from school riding her bike at 20 mph. Part way home she hit a pothole
    and got a flat tire. So she walked her bike the rest of the way home at 2 mph. It took her
    an hour and 30 minutes to get home. If she lives 9 miles from school, for how many
    minutes did she ride her bike?




                                                13
Algebra I – Turner        Classwork – 7




                     14
Algebra I – Turner                                                            Classwork – 7

39. The steamship Empress Anne sailing due West at 32 knots passed the freighter Oregon
    which was sailing due East at 24 knots. In how many hours after the meeting will the
    ships be 448 nautical miles apart?




Lesson Goal: Solve word problems involving mixtures using a single equation or a
system of linear equations.

40. How many grams of water must be added to 40 grams of a 90% acid solution to produce
    a 50% acid solution?




41. How many kilograms of antifreeze must be added to 4 kilograms of a 10% antifreeze
    solution to produce a 20% solution?




                                            15
Algebra I – Turner                                                          Classwork – 7

42. How many grams of water must be evaporated from 10 grams of a 40% antiseptic
    solution to produce a 50% solution?




43. Cream is 30% butterfat and milk is 4% butterfat. How much of each must a dairy mix
    together to get 100 quarts of a product that is 15% butterfat?




                                           16
Algebra I – Turner                                                                       Classwork – 7

Lesson Goal: Solve a word problem using a graph of a system of equations.

44. Suppose x represents the number of people who attend the Dapper Dan basketball game in
    Pittsburgh, Pa. Let C( x)  5x  40,000 represent the cost of hosting the game and
    R( x)  30x represent the total revenue taken in from the game.
                                                                       y
     a.    Graph the functions on Fathom. Use the graph to
           answer the questions below.                           100,000

                                                                  90,000

                                                                  80,000

                                                                  70,000

                                                                  60,000

                                                                  50,000
     b.    For what attendance is revenue equal to operating
                                                                  40,000
           costs? This is called the break-even point.
                                                                  30,000

                                                                  20,000

     c.    For what attendance is revenue greater than            10,000
           operating costs?                                                                                        x
                                                                       0   400 800 1200 1600 2000 2400 2800 3200




     d.    What is the attendance when the revenue is less than operating costs, and how does
           that relate to profit?




     e.    Write an equation or inequality that relates to each question above.




                                                 17
Algebra I – Turner                                                                   Classwork – 7

45. In the United States whole milk consumption was 103 quarts annually per person in 1970
    and fell by 3 quarts per person each year for the next 20 years. By contrast, low fat milk
    consumption was only 25 quarts annually per person in 1970, but rose by 2.25 quarts per
    person each year for the next 20 years. Let x represent the year, with x  0
    corresponding to 1970.

     a.    Write the linear function representing the consumption of whole milk.




     b.    Write the linear function representing the consumption of low fat milk.




     c.    Graph the two functions on Fathom.                  y

                                                         110
                                                         100
                                                         90
     d.    Write an equation that could be used to
           determine in what year the consumption of     80
           whole milk was the same as the                70
           consumption of low fat milk.
                                                         60
                                                         50
                                                         40
                                                         30
     e.    Use the graph to solve the equation.           20
                                                          10
                                                                                                     x
                                                              0    2   4   6   8 1 12 14 16 18 20
                                                                                 0
     f.    Write an inequality that could be used to determine in what years the consumption of
           whole milk was less than the consumption of low fat milk.




     g.    Use the graph to solve the inequality.




                                                    18
Algebra I – Turner                                                                  Classwork – 7

Lesson Goal: Use a graph of two functions to solve a related equation or inequality.

46. A worm starts at the oak tree and moves away, heading for the elm tree at a constant rate of
    13 meters per hour. At the same time a snail starts at the elm tree and moves toward the
    oak tree at a constant rate of 17 m/h. The two trees are 100 m apart. Let x be the
    number of hours the two creatures have been creeping.

     a.    Draw a picture that illustrates what is happening in the problem.




     b.    Write a function f (x) for the distance the worm is from the OAK tree in terms of x.



     c.    Write a function g (x ) for the distance the snail is from the OAK tree in terms of x.



     d.    Graph the two functions on Fathom.

     e.    Write an equation that could be used to answer the question "When do the worm and
           the snail meet?"



     f.    Use the graph to solve the equation.

     g.    Write an inequality that could be solved to answer the question "When is the worm
           farther from the OAK tree than the snail is?"



     h.    Use the graph to solve the inequality.




                                                    19
Algebra I – Turner                                                                  Classwork – 7

47. A company has two tanks of water. One has a capacity of 2000 liters, but now has only
    550 liters of water in it. The other tank has a capacity of 1600 liters and is now full.
    Suppose the company starts to pump water from the 1600 liter tank into the 2000 liter
    tank at a rate of 50 liters per hour. Let x be the amount of time elapsed.

     a.    Write a function for the amount of water in the 2000-liter tank in terms of x.



     b.    Write a function for the amount of water in the 1600-liter tank in terms of x.



     c.    Graph the two functions on Fathom.

     d.    Write an equation that could be used to determine when the two tanks have the same
           amount of water.



     e.    Use the graph to solve the equation.



     f.    Write an inequality that could be used to determine when the 2000-liter tank will
           contain more water than the 1600-liter tank.



     g.    Use the graph to solve the inequality.



     h.    Write an inequality that could be used to determine when the 2000-liter tank will be
           overflowing.



     i.    Use the graph to solve the inequality.




                                                    20
Algebra I – Turner                                                                   Classwork – 7

48. A cougar is lying down beside a tree and spots a fawn 132 meters away. The cougar starts
    running toward the fawn at a speed of 18 meters per second. At the same instant, the fawn
    starts running away at 11 meters per second. Let x be the number of seconds they have
    been running.

     a.    Draw a picture that illustrates what is happening in the problem.




     b.    Write a function f (x) for the distance the cougar is from the tree in terms of x.



     c.    Write a function g (x ) for the distance the fawn is from the tree in terms of x.



     d.    Graph the two functions on Fathom.



     e.    Write an equation that could be used to determine when the cougar catches the fawn.



     f.    Use the graph to solve the equation.



     g.    Write an inequality that could be used to determine when the fawn is farther from the
           tree than the cougar.



     h.    Use the graph to solve the inequality.




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