6-1 Solving Systems by Graphing

[Pages:9]6-1

Solving Systems by Graphing

Vocabulary

Review

Write I if the amount described is infinite. Write F if the amount is finite.

1. the rational numbers greater than 6

2. the number of seats in a movie theater

3. the number of grams in one kilogram

4. the set of odd numbers

5. Give one example of an infinite amount. Explain why the amount is infinite.

You can mod_e_l_t_h__e__p_r_o_b__le__m__I_n__t_h__e__S_o_l_v_e__I_t_w__i_th__t_w__o__l_in__e_a_r__e_q__u_a_t_i_o_n__s_. _T__w__o__o_r_m__ore linear equations form a system of linear equations. Any ordered pair that makes all of the equations in a system true is a solution of a system of lin_e_a_r__e_q__u_a_t_i_o_n__s_.________________________________________________________

You can useVosycsatbeumlasroyfBluiniledaerrequations to model problems. Systems of equations can be solved in more than one way. One method is to graph each equation and find the intersection point, if one exists.

system (noun) SIS tum

PROBLEM 1O:thSeOr WLVoIrNd FGorAmsS:YsySsTteEmMatiOc (FadEjQecUtivAeT),IsOysNteSmBatYizeG(RveArbP)HING

Find

the

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solutions.

linear equations.

A

solution

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7 It C 47 t 0 Complete each statement with the appropriate word from the list.

systematic

system

systematize

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my b 2

M T 6. The librarian planned to 9 the donated magazines.

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7. The American 9 of government is based on the Constitution.

2,4 8. Sam was 9 in his approach to studying for his final exam.

9. A 9 of linear equations might consist of two equations.

a

C2,0

Copyright ? by Pearson Education, Inc. or its affiliates. All Rights Reserved.

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PROBLEM 2: WRITING A SYSTEM OF EQUATIONS

5. Scientists studied the weights of two alligators over a period of 12 months. The initial weight and growth rate of each alligator are shown below. After how many months did the alligators weight the same amount?

Let w weight

m

months

m

w 1 5m74

w mt6

Gator Growth

28

7

24

so

Weight w

lb

12

8

4,10

4

y

I o 2

8 10 12 14 am

months

6. One satellite radio service charges $10 per month plus an activation fee of $20. A second service charges $11 per month plus an activation fee of $15. In what month was the cost of the service the same?

Let E total cost

M months q o

k 10Mt 20 Ilm t 15

m 4 48 400

Total

cost

5t

Sat RadioCosts

7

o

no

e

too

so

w

5,70

40

a

o 2 4 6 8 lo is 14 a months

7. At a local fitness center, members pay a $20 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the same?

Let

total cost

c classes

m ZxY8

3C 1 20 SC

Total

Cost

m fx

es

of AerobicsCosts

so

7

60

so

Cioso

40

so

20

w

b z y b s w iz 14 c

classes

A system of equations that has at least one solution is consistent. A consistent system can be either independent or dependent.

A consistent system that is independent has exactly one solution. For example, the systems in problems 1 through 7 are consistent and independent. A consistent system that is dependent has infinitely many solutions.

A system of equations that has no solution is inconsistent.

PROBLEM 3: SYSTEMS WITH INFINITELY MAY SOLUTIONS OR NO SOLUTION Solve each system by graphing.

1 Eff II 2" - $ = 2

8. 1" = + $ + 1 2

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10. Before graphing the equations, how can you determine whether a system of equations has exactly one solution, infinitely many solutions, or no solution?

One solution

solutions

no solution

diff slopes

g same slope k same intercept

y same slope diff intercepts

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Name __________________________________________

6.1 Practice Worksheet

Is (-1,5) a solution of each system? 1. y = 2x + 7

y = x+6

y = -x + 4 2. y = - 1 x

5

Period ________________

Solve by graphing. Check you solution.

3.

y

=

1 2

x +1

y = -3x + 8

4. 3x + 4y = 12 2x+4y =8

5.

y

=

1 2

x+2

x+y=5

6. y = 4x + 80 -x + 2y = 20

x + 2y = 10 7. 2x + 4y = 12

8. y = 3x + 4 -12x + 4y = 16

Without graphing, decide whether each system has one solution, no solution, or infinitely many

solutions.

9. y = 2x y = 2x-5

10. y = -3x +1 y = 3x + 7

3x - 5y = 0 11. y = 3 x

5

12. Suppose you have $20 in you bank account. You start saving $5 each week. Your friend has $5 in his bank account and is saving $10 each week. Assume that neither you nor your friend makes any withdrawals.

a.) After how many weeks will you and your friend have the same amount of money in your accounts?

b.) How much money will each of you have?

13. Two hikers are walking along a marked trail. The first hiker starts at a point 6 miles from the beginning of the trail and walks at a speed of 3 mi/h. At the same time, the second hiker starts 1 mile from the beginning and walks at a speed of 5 mi/h. Write a system of linear equations and graph them to find when the second hiker will pass the first hiker.

14. y = -5x + w y = -5x + v

a)For what values of w and v does the system have no solution? b)For what values of w and v does the system have infinitely many solutions? c)For what values of w and v does the system have exactly one solution?

15. y = gx + 3 y = hx + 7

a)If ! #, the system has no solution: always, sometimes, or never? b)If ! #, the system has infinitely many solutions: always, sometimes, or never?

16. The slope of the line joining point P to the origin is 2 . The slope of the line joining point P to (-4,3) 9

is 1. Find the coordinates of point P.

17. Which value of b will make the graphs of y = 2x + 3 and y = 2.5x + b intersect at (2,7)?

a) 2

b) 3

c) 5

d) 7

18.

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