Download The Addition Method

7.3
7.3
In this
section
●
Solving a System of Linear
Equations by Addition
●
Inconsistent and
Dependent Systems
●
Applications
E X A M P L E
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383
THE ADDITION METHOD
In Section 7.2 we solved systems of equations by using substitution. We substituted one equation into the other to eliminate a variable. The addition method of
this section is another method for eliminating a variable to solve a system of
equations.
Solving a System of Linear Equations by Addition
In the substitution method we solve for one variable in terms of the other variable.
When doing this, we may get an expression involving fractions, which must be substituted into the other equation. The addition method avoids fractions and is easier
to use on certain systems.
1
Solving a system by addition
Solve:
3x y 5
2x y 10
Solution
The addition property of equality allows us to add the same number to each side of
an equation. If we assume that x and y are numbers that satisfy 3x y 5, then
adding these equations is equivalent to adding 5 to each side of 2x y 10:
calculator
close-up
To check Example 1, graph
y1 3x 5 and y2 10 2x.
The lines appear to intersect
at (3, 4).
3x y 5
2x y 10
5x
15
x3
Add.
y y 0
Note that the y-term was eliminated when we added the equations because the
coefficients of y in the two equations were opposites. Now use x 3 in either one
of the original equations to find y:
10
–10
The Addition Method
10
–10
2x y 10
2(3) y 10
y4
Let x 3.
Check that (3, 4) satisfies both equations. The solution to the system is (3, 4).
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The addition method is based on the addition property of equality. We are
adding equal quantities to each side of an equation. The form of the equations does
not matter as long as the equal signs and the like terms are in line.
In Example 1, y was eliminated by the addition because the coefficients of y in
the two equations were opposites. If no variable will be eliminated by addition, we
can use the multiplication property of equality to change the coefficients of the variables. In the next example the coefficient of x in one equation is a multiple of the
coefficient of x in the other equation. We use the multiplication property of equality
to get opposite coefficients for x.
384
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Chapter 7
E X A M P L E
2
Systems of Linear Equations and Inequalities
Solving a system by addition
Solve:
x 4y 14
2x 3y 18
Solution
If we add these equations as they are written, we will not eliminate any variables.
However, if we multiply each side of the first equation by 2, then we will be adding
2x and 2x, and x will be eliminated:
2(x 4y) 2(14)
2x 3y 18
2x 8y 28
2x 3y 18
5y 10
y 2
Multiply each side by 2.
Add.
Now replace y by 2 in one of the original equations:
x 4(2) 14
x 8 14
x 6
x6
Check x 6 and y 2 in the original equations.
6 4(2) 14
2(6) 3(2) 18
The solution to the system is (6, 2).
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In the next example we need to use a multiple of each equation to eliminate a
variable by addition.
E X A M P L E
study
3
tip
Read carefully. Ask yourself
questions and look for the answers. Every sentence says
something pertaining to the
subject. You cannot read a
mathematics textbook like
you read a novel. You can read
a novel passively, but a textbook requires more concentration and retention.
Solving a system by addition
Solve:
2x 3y 7
3x 4y 10
Solution
To eliminate x by addition, the coefficients of x in the two equations must be opposites. So we multiply the first equation by 3 and the second by 2:
3(2x 3y) 3(7)
2(3x 4y) 2(10)
6x 9y 21
6x 8y 20
y 1
y1
Add.
7.3
The Addition Method
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385
Replace y with 1 in one of the original equations:
2x 3y 7
2x 3(1) 7
2x 3 7
2x 4
x2
Check x 2 and y 1 in the original equations.
2(2) 3(1) 7
3(2) 4(1) 10
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The solution to the system is (2, 1).
If the equations have fractions, you can multiply each equation by the LCD
to eliminate the fractions. Once the fractions are cleared, it is easier to see how to
eliminate a variable by addition.
E X A M P L E
4
A system involving fractions
2
1
Solve:
x y 2
2
3
1
1
x y 6
4
2
Solution
Multiply the first equation by 6 and the second by 4 to eliminate the fractions:
1
2
6 x y 6 2
2
3
1
1
4 x y 4 6
4
2
3x 4y 12
x 2y 24
Now multiply x 2y 24 by 2 to get 2x 4y 48, and then add:
3x 4y 12
2x 4y 48
5x
60
x 12
Let x 12 in x 2y 24:
12 2y 24
2y 12
y6
Check x 12 and y 6 in the original equations. The solution is (12, 6).
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386
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Chapter 7
Systems of Linear Equations and Inequalities
Use the following strategy to solve a system by addition.
Strategy for Solving a System by Addition
1. Write both equations in standard form.
2. If a variable will be eliminated by adding, then add the equations.
3. If necessary, obtain multiples of one or both equations so that a variable will
be eliminated by adding the equations.
4. After one variable is eliminated, solve for the remaining variable.
5. Use the value of the remaining variable to find the value of the eliminated
variable.
6. Check the solution in the original system.
Inconsistent and Dependent Systems
When the addition method is used, an inconsistent system will be indicated by
a false statement. A dependent system will be indicated by an equation that is
always true.
E X A M P L E
5
Inconsistent and dependent systems
Use the addition method to solve each system.
a) 2x 3y 9
b) 2x y 1
2x 3y 18
4x 2y 2
Solution
a) Add the equations:
2x 3y 9
2x 3y 18
0 27
False.
There is no solution to the system. The system is inconsistent.
b) Multiply the first equation by 2, and then add the equations:
2(2x y) 2(1)
4x 2y 2
4x 2y 2
4x 2y 2
00
True.
Because the equation 0 0 is correct for any value of x, the system is depen■
dent. The set of points satisfying the system is (x, y ) 2x y 1.
Applications
In the next example we solve a problem using a system of equations and the
addition method.
7.3
E X A M P L E
helpful
6
hint
You can see from Example 6,
that the standard form Ax By C occurs naturally in accounting. This form will occur
whenever we have the price
each and quantity of two
items and we want to express
the total cost.
The Addition Method
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387
Milk and bread
Lea purchased two gallons of milk and three loaves of bread for $8.25. Yesterday
she purchased five gallons of milk and two loaves of bread for $13.75. What is the
price of a single gallon of milk? What is the price of a single loaf of bread?
Solution
Let x represent the price of one gallon of milk. Let y represent the price of one loaf
of bread. We can write two equations about the milk and bread:
2x 3y 8.25
5x 2y 13.75
Today’s purchase
Yesterday’s purchase
To eliminate x, multiply the first equation by 5 and the second by 2:
5(2x 3y) 5(8.25)
2(5x 2y) 2(13.75)
10x 15y 41.25
10x 4y 27.50
Add.
11y 13.75
y 1.25
Replace y by 1.25 in the first equation:
2x 3(1.25) 8.25
2x 3.75 8.25
2x 4.50
x 2.25
A gallon of milk costs $2.25, and a loaf of bread costs $1.25.
WARM-UPS
True or false? Explain your answer.
Use the following systems for these exercises:
a) 3x 2y 7
b) y 3x 2
c) y x 5
4x 5y 6
2y 6x 4 0
xy6
1. To eliminate x by addition in system (a), we multiply the first equation by 4
and the second equation by 3. False
2. Either variable in system (a) can be eliminated by the addition method.
True
3. The ordered pair (1, 2) is a solution to system (a). True
4. The addition method can be used to eliminate a variable in system (b).
True
5. Both (0, 2) and (1, 1) satisfy system (b). True
6. The solution to system (c) is (x, y) y x 5. False
7. System (c) is independent. False
8. System (b) is inconsistent. False
9. System (a) is dependent. False
10. The graphs of the equations in system (c) are parallel lines. True
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7.3
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Chapter 7
Systems of Linear Equations and Inequalities
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What method is used in this section to solve systems of
equations?
In this section we learned to solve systems by the addition
method.
2. What three methods have now been presented for solving a
system of linear equations?
The three methods presented are graphing, substitution,
and addition.
3. What do the addition method and the substitution method
have in common?
In addition and substitution we eliminate a variable and
solve for the remaining variable.
4. What do we sometimes do before we add the equations?
It is sometimes necessary to use the multiplication property
of equality before adding the equations.
5. How do you decide which variable to eliminate when using
the addition method?
Eliminate the variable that is easiest to eliminate.
6. How do you identify inconsistent and dependent systems
when using the addition method?
In the addition method inconsistent systems result in a false
equation and dependent systems result in an equation that is
always true.
Solve each system by the addition method. See Examples 1–4.
7. 2x y 5
8. 3x y 3
3x y 10 (3, 1)
4x y 11 (2, 3)
9. x 2y 7
x 3y 18
11. x 2y 2
4x 3y 25
13. x 3y 4
2x y 1
15. y 4x 1
y 3x 7
(3, 5)
(4, 3)
1 9
, 7 7
(8, 31)
10. x 2y 7
x 4y 5
(3, 2)
12. 2x 3y 7
5x y 9 (2, 1)
14. x y 0
x 2y 0
16. x 3y 45
x 2y 40
(0, 0)
(30, 5)
Use either the addition method or substitution to solve each
system. State whether the system is independent, inconsistent,
or dependent. See Example 5.
27. x y 5
x y 6 No solution, inconsistent
28. x y 5
x 2y 6
(4, 1), independent
29. x y 5
2x 2y 10
(x, y) x y 5, dependent
30. 2x 3y 4
2x 3y 4 (2, 0), independent
31. 2x y 3
2y 4x 6
(x, y) 2x y 3, dependent
32. y 2x 1
2x y 5 0
No solution, inconsistent
33. x 3y 3
5x 15 15y (x, y) x 3y 3, dependent
34. y 3x 2
5y 15x 10
(0, 2), independent
35. 6x 2y 2
1
4
y x No solution, inconsistent
3
3
36. x y 8
1
1
x y 1 (6, 2), independent
3
2
1
2
37. x y 6
2
3
3
1
x y 18 (12, 18), independent
4
2
1
38. x y 3
2
1
x 2y 6 (10, 2), independent
5
39. 0.04x 0.09y 7
x y 100 (40, 60), independent
40. 0.08x 0.05y 0.2
2x y 140 (40, 60), independent
17. 4x 3y 1
2x y 1 (2, 3)
18. 2x y 9
x 1 3y (4, 1)
19. 2x 5y 22
6x 3y 18
(1, 4)
20. 4x 3y 7
5x 6y 1 (1, 1)
42. 0.5y 0.2x 0.25
0.1y 0.8x 1.57 (2, 0.3), independent
21. 2x 3y 4
3x 5y 13
(1, 2)
22. 5x 3y 1
2x 7y 17 (2, 3)
Use a calculator to assist you in finding the exact solution to
each system.
24. 4x 3y 17
3x 5y 21
(2, 3)
43. 2.33x 4.58y 16.319
4.98x 3.44y 2.162
(1, 4)
44. 234x 499y 1337
282x 312y 51,846
23. 2x 5y 11
3x 2y 11
(3, 1)
25. 5x 4y 13
2x 3y 8 (1, 2)
26. 4x 3y 8
6x 5y 14
41. 0.1x 0.2y 0.01
0.3x 0.5y 0.08 (0.1, 0.1), independent
(1.5, 2.8)
(123, 55)
7.3
Use two variables and a system of equations to solve each problem. See Example 6.
45. Cars and trucks. An automobile dealer had 250 vehicles
on his lot during the month of June. He must pay a monthly
inventory tax of $3 per car and $4 per truck. If his tax bill
for June was $850, then how many cars and how many
trucks did he have on his lot during June?
150 cars, 100 trucks
FIGURE FOR EXERCISE 45
46. Dimes and nickels. Kimberly opened a parking meter and
removed 30 coins consisting of dimes and nickels. If the
value of these coins is $2.30, then how many of each type
does she have?
16 dimes, 14 nickels
47. Adults and children. The Audubon Zoo charges $5.50 for
each adult admission and $2.75 for each child. The total bill
for the 30 people on the Spring Creek Elementary School
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kindergarten field trip was $99. How many adults and how
many children went on the field trip?
6 adults, 24 children
48. Coffee and doughnuts. Jorge has worked at Dandy
Doughnuts so long that he has memorized the amounts for
many of the common orders. For example, six doughnuts
and five coffees cost $4.35, while four doughnuts and three
coffees cost $2.75. What are the prices of one cup of coffee
and one doughnut?
Coffee $0.45, doughnut $0.35
49. Marketing research. The Independent Marketing Research Corporation found 130 smokers among 300 adults
surveyed. If one-half of the men and one-third of the
women were smokers, then how many men and how many
women were in the survey?
180 men, 120 women
50. Time and a half. In one month, Shelly earned $1800 for
210 hours of work. If she earns $8 per hour for regular time
and $12 per hour for overtime, then how many hours of
each type did she work?
180 hours regular time, 30 hours overtime
GET TING MORE INVOLVED
51. Discussion. Compare and contrast the three methods for
solving systems of linear equations in two variables that
were presented in this chapter. What are the advantages and
disadvantages of each method? How do you choose which
method to use?
52. Exploration. Consider the following system:
a1x b1y c1
a2 x b2y c2
a) Multiply the first equation by a2 and the second equation by a1. Add the resulting equations and solve for y
to get a formula for y in terms of the a’s, b’s, and c’s.
a a b
b) Multiply the first equation by b2 and the second by b1.
Add the resulting equations and solve for x to get a
formula for x in terms of the a’s, b’s, and c’s.
bc b
x b ba
c) Use the formulas that you found in (a) and (b) to find the
solution to the following system:
2x 3y 7
5x 4y 14
FIGURE FOR EXERCISE 47
(2, 1)