6-3 Elimination Using Addition and Subtraction
Use elimination to solve each system of equations.
1. 5m − p = 7
7m − p = 11
SOLUTION: Multiply the second equation by −1 .
Then, add this to the first equation.
Now, substitute 2 for m in either equation to find the value of p .
The solution is (2, 3).
Check the solution in both equations.
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Page 1
The solution Using
is (2, 3).
6-3 Elimination
Addition and Subtraction
Check the solution in both equations.
2. 8x + 5y = 38
−8x + 2y = 4
SOLUTION: Because −8x and 8x have opposite coefficients, add the equations.
Now, substitute 6 for y in either equation to find the value of x.
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Page 2
6-3 Elimination
Using Addition and Subtraction
Now, substitute 6 for y in either equation to find the value of x.
The solution is (1, 6).
Check the solution in both equations. 3. 7f + 3g = −6
7f − 2g = −31
SOLUTION: Because 7f and 7f have the same coefficients, subtract the equations.
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Page 3
3. 7f + 3g = −6
7f − 2g = −31
SOLUTION: 6-3 Elimination
Using Addition and Subtraction
Because 7f and 7f have the same coefficients, subtract the equations.
Now, substitute 5 for g in either equation to find the value of f .
The solution is (−3, 5).
Check the solution in both equations. eSolutions Manual - Powered by Cognero
4. 6a − 3b = 27
Page 4
6-3 Elimination Using Addition and Subtraction
4. 6a − 3b = 27
2a − 3b = 11
SOLUTION: Because 3b and 3b have the same coefficients, multiply the second equation by –1, and add the equations to solve for a. Now, substitute 4 for a in either equation to find the value of b.
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The solution is (4, −1).
Check the solution in each equation.
Page 5
6-3 Elimination
Using Addition and Subtraction
Now, substitute 4 for a in either equation to find the value of b.
The solution is (4, −1).
Check the solution in each equation.
5. CCSS REASONING The sum of two numbers is 24. Five times the first number minus the second number is 12. What are the two numbers?
SOLUTION: Let x represent one number and y represent the second number.
x + y = 24
5x – y = 12
Because y and −y have opposite coefficients, add the equations.
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Page 6
SOLUTION: Let x represent one number and y represent the second number.
x + y = 24
5x – y = 12 Using Addition and Subtraction
6-3 Elimination
Because y and −y have opposite coefficients, add the equations.
Now, substitute 6 for x in either equation to find the value of y.
The two numbers are 6 and 18.
Check the numbers in each equation.
6. RECYCLING The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they
provide 1,275,000 jobs. How many jobs does each industry provide?
SOLUTION: Let y represent the number of recycling workers and let x represent the number of waste management workers.
x + 1,025,000 = y
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x + y = 1,275,000
Rearrange the first equation to get the variables on the same side.
Page 7
6. RECYCLING The recycling and reuse industry employs approximately 1,025,000 more workers than the waste management industry. Together they
provide 1,275,000 jobs. How many jobs does each industry provide?
6-3 Elimination Using Addition and Subtraction
SOLUTION: Let y represent the number of recycling workers and let x represent the number of waste management workers.
x + 1,025,000 = y
x + y = 1,275,000
Rearrange the first equation to get the variables on the same side.
Because x and x have the same coefficients, subtract the two equations.
Now, substitute 1,150,000 for y in either equation to find the value of x.
There are 125,000 waste management workers and 1,150,000 recycling workers.
Check the solution in each equation.
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Page 8
There are 125,000
management
workers and 1,150,000 recycling workers.
6-3 Elimination
Usingwaste
Addition
and Subtraction
Check the solution in each equation.
Use elimination to solve each system of equations.
7. −v + w = 7
v+w=1
SOLUTION: Because −v and v have opposite coefficients, add the equations.
Now, substitute 4 for w in either equation to find the value of v.
The solution is (−3, 4).
Check the solution in each equation.
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8. y + z = 4
Page 9
6-3 Elimination Using Addition and Subtraction
Use elimination to solve each system of equations.
7. −v + w = 7
v+w=1
SOLUTION: Because −v and v have opposite coefficients, add the equations.
Now, substitute 4 for w in either equation to find the value of v.
The solution is (−3, 4).
Check the solution in each equation.
8. y + z = 4
y −z = 8
SOLUTION: Because z and −z have opposite coefficients, add the equations.
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Page 10
6-38. y
Elimination
Using Addition and Subtraction
+z = 4
y −z = 8
SOLUTION: Because z and −z have opposite coefficients, add the equations.
Now, substitute 6 for y in either equation to find the value of z.
The solution is (6, −2).
Check the solution in each equation.
9. −4x + 5y = 17
4x + 6y = −6
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SOLUTION: Because 4x and −4x have opposite coefficients, add the equations.
Page 11
6-3 Elimination Using Addition and Subtraction
9. −4x + 5y = 17
4x + 6y = −6
SOLUTION: Because 4x and −4x have opposite coefficients, add the equations.
Now, substitute 1 for y in either equation to find the value of x.
The solution is (−3, 1).
Check the solution in each equation.
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Page 12
6-3 Elimination Using Addition and Subtraction
10. 5m − 2p = 24
3m + 2p = 24
SOLUTION: Because 2p and −2p have opposite coefficients, add the equations.
Now, substitute 6 for m in either equation to find the value of p .
The solution is (6, 3).
Check the solution in each equation.
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Page 13
The solution Using
is (6, 3).
6-3 Elimination
Addition and Subtraction
Check the solution in each equation.
11. a + 4b = −4
a + 10b = −16
SOLUTION: Because a and a have the same coefficients, subtract the equations.
Now, substitute −2 for b in either equation to find the value of a.
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by Cognero
The Manual
solution
is (4, −2).
Check the solution in each equation.
Page 14
6-3 Elimination Using Addition and Subtraction
The solution is (4, −2).
Check the solution in each equation.
12. 6r − 6t = 6
3r − 6t = 15
SOLUTION: Because −6t and −6t have the same coefficients, multiply equation 2 by –1 and then add the equations to solve for r.
Now, substitute −3 for r in either equation to find the value of t.
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Page 15
6-3 Elimination
Using Addition and Subtraction
Now, substitute −3 for r in either equation to find the value of t.
The solution is (−3, −4).
Check the solution in each equation.
13. 6c − 9d = 111
5c − 9d = 103
SOLUTION: Because −9d and −9d have the same coefficients, subtract the equations.
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Now, substitute 8 for c in either equation to find the value of d.
Page 16
13. 6c − 9d = 111
5c − 9d = 103
SOLUTION: 6-3 Elimination Using Addition and Subtraction
Because −9d and −9d have the same coefficients, subtract the equations.
Now, substitute 8 for c in either equation to find the value of d.
The solution is (8, −7).
Check the solution in each equation.
14. 11f + 14g = 13
11f + 10g = 25
SOLUTION: eSolutions
Manual - Powered by Cognero
Because 11f and 11f have the same coefficients, you can multiply equation 2 by −1, then add the equations to find g.
Page 17
6-3
Elimination
Using Addition and Subtraction
14. 11f
+ 14g = 13
11f + 10g = 25
SOLUTION: Because 11f and 11f have the same coefficients, you can multiply equation 2 by −1, then add the equations to find g.
Now, substitute −3 for g in either equation to find the value of f .
The solution is (5, −3).
Check the solution in each each equation.
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Page 18
The solution is
(5, −3).
6-3 Elimination
Using
Addition and Subtraction
Check the solution in each each equation.
15. 9x + 6y = 78
3x − 6y = −30
SOLUTION: Because 6y and −6y have opposite coefficients, add the equations.
Now, substitute 4 for x in either equation to find the value of y.
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Page 19
6-3 Elimination
Using Addition and Subtraction
Now, substitute 4 for x in either equation to find the value of y.
The solution is (4, 7).
Check the solution in each equation.
16. 3j + 4k = 23.5
8j − 4k = 4
SOLUTION: Because 4k and −4k have opposite coefficients, add the equations.
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Page 20
16. 3j + 4k = 23.5
8j − 4k = 4
SOLUTION: 6-3 Elimination Using Addition and Subtraction
Because 4k and −4k have opposite coefficients, add the equations.
Now, substitute 2.5 for j in either equation to find the value of k .
The solution is (2.5, 4).
Check the solution in both equations.
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17. −3x − 8y = −24
3x − 5y = 4.5
Page 21
6-3 Elimination Using Addition and Subtraction
17. −3x − 8y = −24
3x − 5y = 4.5
SOLUTION: Because −3x and 3x have opposite coefficients, add the equations.
Now, substitute 1.5 for y in either equation to find the value of x.
The solution is (4, 1.5).
Check the solution in each equation.
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Page 22
The solution Using
is (4, 1.5).
6-3 Elimination
Addition and Subtraction
Check the solution in each equation.
18. 6x − 2y = 1
10x − 2y = 5
SOLUTION: Because −2y and −2y have the same coefficients, subtract the equations.
Now, substitute 1 for x in either equation to find the value of y.
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Page 23
6-3 Elimination
Using Addition and Subtraction
Now, substitute 1 for x in either equation to find the value of y.
The solution is (1, 2.5).
Check the solution in each equation.
19. The sum of two numbers is 22, and their difference is 12. What are the numbers?
SOLUTION: Let x represent one number and y represent the second number.
Because y and −y have opposite coefficients, add the equations.
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Page 24
6-3 Elimination Using Addition and Subtraction
19. The sum of two numbers is 22, and their difference is 12. What are the numbers?
SOLUTION: Let x represent one number and y represent the second number.
Because y and −y have opposite coefficients, add the equations.
Now, substitute 17 for x in either equation to find the value of y.
The two numbers are 17 and 5.
20. Find the two numbers with a sum of 41 and a difference of 9.
SOLUTION: Let x represent one number and y represent the second number.
Because y and −y have opposite coefficients, add the equations.
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Page 25
6-3 Elimination Using Addition and Subtraction
The two numbers are 17 and 5.
20. Find the two numbers with a sum of 41 and a difference of 9.
SOLUTION: Let x represent one number and y represent the second number.
Because y and −y have opposite coefficients, add the equations.
Now, substitute 25 for x in either equation to find the value of y.
The two numbers are 25 and 16.
21. Three times a number minus another number is −3. The sum of the numbers is 11. Find the numbers.
SOLUTION: Let x represent one number and y represent the second number.
Because −y and y have opposite coefficients, add the equations.
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Page 26
6-3 Elimination
Using Addition and Subtraction
The two numbers are 25 and 16.
21. Three times a number minus another number is −3. The sum of the numbers is 11. Find the numbers.
SOLUTION: Let x represent one number and y represent the second number.
Because −y and y have opposite coefficients, add the equations.
Now, substitute 2 for x in either equation to find the value of y.
The two numbers are 2 and 9.
22. A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?
SOLUTION: Let x represent one number and y represent the second number.
Because −2y and 2y have opposite coefficients, add the equations.
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Page 27
6-3 Elimination
Using Addition and Subtraction
The two numbers are 2 and 9.
22. A number minus twice another number is 4. Three times the first number plus two times the second number is 12. What are the numbers?
SOLUTION: Let x represent one number and y represent the second number.
Because −2y and 2y have opposite coefficients, add the equations.
Now, substitute 4 for x in either equation to find the value of y.
The two numbers are 4 and 0.
23. TOURS The Blackwells and Joneses are going to Hershey’s Really Big 3D Show in Pennsylvania. Find the adult price and the children’s price of the show.
SOLUTION: Let x represent the number of adult tickets and y represent the number of children’s tickets.
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Page 28
6-3 Elimination Using Addition and Subtraction
SOLUTION: Let x represent the number of adult tickets and y represent the number of children’s tickets.
Because 2x and 2x have the same coefficients, multiply equation 2 by –1, then add the equations to find y.
Now, substitute 3.95 for y in either equation to find the value of x.
The children’s tickets are $3.95 and the adult tickets are $5.95.
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Use elimination to solve each system of equations.
24. 4(x + 2y) = 8
4x + 4y = 12
Page 29
6-3 Elimination
Using Addition and Subtraction
The children’s tickets are $3.95 and the adult tickets are $5.95.
Use elimination to solve each system of equations.
24. 4(x + 2y) = 8
4x + 4y = 12
SOLUTION: Distribute the 4 in the first equation.
Because 4x and 4x have the same coefficients, subtract the equations.
Now, substitute −1 for y in either equation to find the value of x.
The solution is (4, −1).
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25. 3x − 5y = 11
5(x + y) = 5
Page 30
6-3 Elimination
Using Addition and Subtraction
The children’s tickets are $3.95 and the adult tickets are $5.95.
Use elimination to solve each system of equations.
24. 4(x + 2y) = 8
4x + 4y = 12
SOLUTION: Distribute the 4 in the first equation.
Because 4x and 4x have the same coefficients, subtract the equations.
Now, substitute −1 for y in either equation to find the value of x.
The solution is (4, −1).
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25. 3x −Manual
5y = 11
5(x + y) = 5
SOLUTION: Page 31
6-3 Elimination
Using
Addition and Subtraction
The solution is
(4, −1).
25. 3x − 5y = 11
5(x + y) = 5
SOLUTION: Distribute the 5 in the second equation.
Because −5y and 5y have opposite coefficients, add the equations.
Now, substitute 2 for x in either equation to find the value of y.
The solution is (2, −1).
26. 4x + 3y = 6
3x + 3y = 7
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SOLUTION: Because 3y and 3y have the same coefficients, multiply equation 2 by −1, and add the equations to find x..
Page 32
6-3 Elimination
Using Addition and Subtraction
The solution is (2, −1).
26. 4x + 3y = 6
3x + 3y = 7
SOLUTION: Because 3y and 3y have the same coefficients, multiply equation 2 by −1, and add the equations to find x..
Now, substitute −1 for x in either equation to find the value of y.
The solution is
.
27. 6x − 7y = −26
6x +Manual
5y = 10
eSolutions
- Powered by Cognero
SOLUTION: Because 6x and 6x have the same coefficients, subtract the equations.
Page 33
6-3 Elimination
Using Addition
and Subtraction
The solution is
.
27. 6x − 7y = −26
6x + 5y = 10
SOLUTION: Because 6x and 6x have the same coefficients, subtract the equations.
Now, substitute 3 for y in either equation to find the value of x.
The solution is
.
28. SOLUTION: eSolutions Manual - Powered by Cognero
Because
y and − y have the same coefficients, add the equations.
Page 34
28. 6-3 Elimination Using Addition and Subtraction
SOLUTION: Because
y and − y have the same coefficients, add the equations.
Now, substitute 12 for x in either equation to find the value of y.
The Manual
solution
is
.
eSolutions
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by Cognero
29. Page 35
6-3 Elimination Using Addition and Subtraction
The solution is
.
29. SOLUTION: Because
x and − x have opposite coefficients, add the equations.
Now, substitute
for y in either equation to find the value of x.
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Page 36
6-3 Elimination
Using Addition
and Subtraction
Now, substitute
for y in either
equation to find the value of x.
The solution is
.
30. CCS SENSE-MAKING The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights
between the building and the statue is 295.4 feet.
a. How tall is the statue?
b. How tall is the building?
SOLUTION: a. Because g and –g have opposite coefficients, add the equations.
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Page 37
6-3 Elimination Using Addition and Subtraction
The solution is
.
30. CCS SENSE-MAKING The total height of an office building b and the granite statue that stands on top of it g is 326.6 feet. The difference in heights
between the building and the statue is 295.4 feet.
a. How tall is the statue?
b. How tall is the building?
SOLUTION: a. Because g and –g have opposite coefficients, add the equations.
Now, substitute 311 for b in either equation to find the value of g.
The height of the statue is 15.6 ft.
b. The height of the building is 311 ft.
31. BIKE RACING Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams.
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38
a. Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number ofPage
U.S.
teams and non-U.S. teams.
The height of the statue is 15.6 ft.
b. The height of the building is 311 ft.
6-3 Elimination Using Addition and Subtraction
31. BIKE RACING Professional Mountain Bike Racing currently has 66 teams. The number of non-U.S. teams is 30 more than the number of U.S. teams.
a. Let x represent the number of non-U.S. teams and y represent the number of U.S. teams. Write a system of equations that represents the number of U.S.
teams and non-U.S. teams.
b. Use elimination to find the solution of the system of equations.
c. Interpret the solution in the context of the situation.
d. Graph the system of equations to check your solution.
SOLUTION: a. The total number of teams is 66, so x + y = 66. There are 30 more of x than y, so x = 30 + y.
b. Rearrange the second equation to get the variables on the same side.
Because y and –y have opposite coefficients, add the equations.
Now, substitute 48 for x in either equation to find the value of y.
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Page 39
6-3 Elimination Using Addition and Subtraction
Now, substitute 48 for x in either equation to find the value of y.
(48, 18)
c. This answer means that there are 48 non-U.S. teams and 18 U.S. Teams in Mountain Bike Racing.
d.
32. SHOPPING Let x represent the number of years since 2004 and y represent the number of catalogs.
a. Write a system of equations to represent this situation.
b. Use elimination to find the solution to the system of equations.
c. Analyze the solution in terms of the situation. Determine the reasonableness of the solution.
SOLUTION: Page 40
a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and
the growth rates represent the slopes. Write the equation for number of each type of catalogs.
online: y = 1293x + 7440
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a. Write a system of equations to represent this situation.
b. Use elimination to find the solution to the system of equations.
6-3 Elimination
Using Addition and Subtraction
c. Analyze the solution in terms of the situation. Determine the reasonableness of the solution.
SOLUTION: a. Since x represents the number of years since 2004, x = 0 would represent the year 2004. Then the numbers of catalogs in 2004 are the y-intercepts and
the growth rates represent the slopes. Write the equation for number of each type of catalogs.
online: y = 1293x + 7440
print: y = –1364x + 3805
b. Since both equations contain y , use elimination. Negate the second equation.
Now use elimination by addition.
Now, substitute −1.36808 for x in either equation to find the value of y.
The solution is about (−1.4, 5671.1)
c. This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems
reasonable.
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Page 41
33. MULTIPLE REPRESENTATIONS Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point
6-3 Elimination
Using Addition and Subtraction
The solution is about (−1.4, 5671.1)
c. This means about 1.4 years before 2004, or in 2002, the number of online catalogs and the number of print catalogs were both 5671. This seems
reasonable.
33. MULTIPLE REPRESENTATIONS Collect 9 pennies and 9 paper clips. For this game, you use 9 objects to score points. Each paper clip is worth 1 point
and each penny is worth 3 points. Let p represent the number of pennies and c represent the number of paper clips.
a. CONCRETE Choose a combination of 9 objects and find your score.
b. ANALYTICAL Write and solve a system of equations to find the number of paper clips and pennies used for 15 points.
c. TABULAR Make a table showing the number of paper clips used and the total number of points when the number of pennies is 0, 1, 2, 3, 4, or 5.
d. VERBAL Does the result in the table match the results in part b? Explain.
SOLUTION: a. Sample answer: If you choose 4 pennies and 5 paper clips, the score will be 4(3) + 5 or 17.
b. Let p represent the number of pennies and c represent the number of paper clips.
p +c=9
3p + c = 15
Since both equations contain c, use elimination by subtraction.
Now, substitute 3 for p in either equation to find the value of c.
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Page 42
6-3 Elimination Using Addition and Subtraction
Now, substitute 3 for p in either equation to find the value of c.
So, the solution is p = 3, c = 6.
c. Sample answer:
d. Yes. Since the pennies are 3 points each, 3 of them makes 9 points. Add the 6 points from 6 paper clips and you get 15 points. Using 9 objects, there is no
other way of obtaining a score of 15 points.
34. REASONING Describe the solution of a system of equations if after you added two equations the result was 0 = 0.
SOLUTION: If the result is a true statement such as 0 = 0, then there would be an infinite number of solutions. A system that has an infinite number of solutions is
consistent and dependent. The two equations represent the same line.
For example, 2x + 5y = 4 and -2x - 5y = -4 when added together give you 0 = 0 as both variables are eliminated. This system would have infinitely
many solutions. When you write each equation in slope-intercept form, you get . Since both equations have the same slope-intercept
form, the equations represent the same line.
35. REASONING What is the solution of a system of equations if the sum of the equations is 0 = 2?
SOLUTION: The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel
lines.
For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.
Each of the lines has a slope of -2. Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have
no points in common. So, there is no solution. eSolutions Manual - Powered by Cognero
36. OPEN ENDED Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such
systems.
Page 43
The result of the statement is false, so there is no solution. This system of equations is inconsistent. The equations in an inconsistent system represent parallel
lines.
For example, the sum of equations 2x + y = 5 and -2x - y = 3 is 0 = 2. When the lines are written in slope-intercept form, you get y = -2x + 5 and y = -2x + 3.
6-3 Elimination
Usinghas
Addition
and
a slope of
-2.Subtraction
Since the lines have the same slopes but different y-intercepts, they are parallel. Parallel lines do not intersect and have
Each of the lines
no points in common. So, there is no solution. 36. OPEN ENDED Create a system of equations that can be solved by using addition to eliminate one variable. Formulate a general rule for creating such
systems.
SOLUTION: Sample answer: Write an equation using two variables, such as 2a + b = 5. Next, write a second equation using a coefficient for one the variables that is the
opposite of the coefficient of that variable in the first equation. Since -b and b have opposite coefficients, the second equation could be a − b = 4.; a system
that can be solved by using addition to eliminate one variable must have one variable with coefficients that are additive inverses (opposites).
37. CCSS STRUCTURE The solution of a system of equations is (−3, 2). One equation in the system is x + 4y = 5. Find a second equation for the system.
Explain how you derived this equation.
SOLUTION: Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of
n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system could be −x + y = 5.; I used the solution to create another equation with the coefficient
of the x-term being opposite of its corresponding coefficient.
38. CHALLENGE The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is –4. Define the variables and
write the system of equations that you would use to find the number. Then solve the system and find the number.
SOLUTION: Let a = the tens digit of the number, and let b = the ones digit of the number.
a – b = –4; a + b = 8
Solve using elimination. Find b. eSolutions Manual - Powered by Cognero
Then the number is 26.
Page 44
SOLUTION: Sample answer: Write an equation using the opposite coefficient for x, such as x + y = n, where n is a number. Use the solution (-3, 2) to find the value of
n. Since -(-3) + 2 = 3 + 2 or 5, a second equation for the system could be 6-3 Elimination
Using Addition and Subtraction
−x + y = 5.; I used the solution to create another equation with the coefficient
of the x-term being opposite of its corresponding coefficient.
38. CHALLENGE The sum of the digits of a two-digit number is 8. The result of subtracting the units digit from the tends digit is –4. Define the variables and
write the system of equations that you would use to find the number. Then solve the system and find the number.
SOLUTION: Let a = the tens digit of the number, and let b = the ones digit of the number.
a – b = –4; a + b = 8
Solve using elimination. Find b. Then the number is 26.
39. WRITING IN MATH Describe when it would be most beneficial to use elimination to solve a system of equations.
SOLUTION: Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite coefficients in the equations. If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps.
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Using Addition and Subtraction
Then the number is 26.
39. WRITING IN MATH Describe when it would be most beneficial to use elimination to solve a system of equations.
SOLUTION: Sample answer: It would be most beneficial when one variable has either the same coefficient or opposite coefficients in the equations. If the system of equations is 3x - 5y = 12 and 2x + 5y = 18, then using elimination by addition to solve the system requires no additional steps.
It is usually not beneficial to use elimination when the equations do not have the like terms aligned on the same sides of the equations. Suppose the system of equations is 2x + 3y = 6 and y = -2x - 14.
Using elimination would first require rewriting the equations so that the variables would align. Then the solution could be found by using elimination by subtraction. Substitution might be a more beneficial method to use in this case.
40. SHORT RESPONSE Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. If the train has been traveling for an hour,
how many more hours is her train ride?
SOLUTION: eSolutions
Manual=- rate
Powered
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Distance
⋅ time
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Substitution might be a more beneficial method to use in this case.
40. SHORT RESPONSE Martina is on a train traveling at a speed of 188 mph between two cities 1128 miles apart. If the train has been traveling for an hour,
how many more hours is her train ride?
SOLUTION: Distance = rate ⋅ time
It will take 6 hours for the total trip. Since she has already traveled for 1 hour, she has 5 more hours to go.
41. GEOMETRY Ms. Miller wants to tile her rectangular kitchen floor. She knows the dimensions of the floor. Which formula should she use to find the area?
A A =
w
B V = Bh
C P = 2
+ 2w
2
2
D c = a + b
2
SOLUTION: Ms. Miller wants to find the area of the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the
Pythagorean Theorem.
42. If the pattern continues, what is the 8th number in the sequence?
F G eSolutions Manual - Powered by Cognero
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2
2
D c = a + b
2
SOLUTION: Ms. Miller wants
find the area
the floor, so she should use the formula in choice A. Choice B is for volume, C is for perimeter, and D is the
6-3 Elimination
UsingtoAddition
andofSubtraction
Pythagorean Theorem.
42. If the pattern continues, what is the 8th number in the sequence?
F G H J SOLUTION: If the pattern continues, the following numbers will be in the series:
.
This means that choice F is correct.
43. What is the solution of this system of equations?
x + 4y = 1
2x − 3y = −9
A (2, −8)
B (−3, 1)
C no solution
D infinitely many solutions
SOLUTION: eSolutions
Manual
- Powered
by Cognero
Solve
the first
equation
for x since
its coefficient is 1.
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C no solution
6-3 Elimination
Usingsolutions
Addition and Subtraction
D infinitely many
SOLUTION: Solve the first equation for x since its coefficient is 1.
Substitute 1 - 4y for x in the second equation to find the value of y .
Now, substitute 1 for y in either equation to find the value of x.
The solution is (−3, 1). Therefore, the correct choice is B.
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or
infinitely many solutions.
44. y = 6x
2x + 3y = 40
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Using Addition and Subtraction
The solution is (−3, 1). Therefore, the correct choice is B.
Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or
infinitely many solutions.
44. y = 6x
2x + 3y = 40
SOLUTION: Now, substitute 2 for x in either equation to find the value of y.
The solution is (2, 12).
45. x = 3y
2x + 3y = 45
SOLUTION: eSolutions Manual - Powered by Cognero
Now, substitute 5 for y in either equation to find the value of x.
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The solution Using
is (2, 12).
45. x = 3y
2x + 3y = 45
SOLUTION: Now, substitute 5 for y in either equation to find the value of x.
The solution is (15, 5).
46. x = 5y + 6
x = 3y − 2
SOLUTION: eSolutions Manual - Powered by Cognero
Now, substitute −4 for y in either equation to find the value of x.
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The solution is (15, 5).
46. x = 5y + 6
x = 3y − 2
SOLUTION: Now, substitute −4 for y in either equation to find the value of x.
The solution is (−14, −4).
47. y = 3x + 2
y = 4x − 1
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Using Addition and Subtraction
The solution is (−14, −4).
47. y = 3x + 2
y = 4x − 1
SOLUTION: Now, substitute 3 for x in either equation to find the value of y.
The solution is (3, 11).
48. 3c = 4d + 2
c=d−1
SOLUTION: eSolutions Manual - Powered by Cognero
Now, substitute −5 for d in either equation to find the value of c.
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Using Addition and Subtraction
The solution is (3, 11).
48. 3c = 4d + 2
c=d−1
SOLUTION: Now, substitute −5 for d in either equation to find the value of c.
The solution is (−6, −5).
49. z = v + 4
2z − v = 6
SOLUTION: Now, substitute −2 for v in either equation to find the value of z.
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6-3 Elimination
Using Addition and Subtraction
The solution is (−6, −5).
49. z = v + 4
2z − v = 6
SOLUTION: Now, substitute −2 for v in either equation to find the value of z.
The solution is (−2, 2).
50. FINANCIAL LITERACY Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has
$26 and plans to save $13 per week.
a. In how many weeks will Gregorio and Javier have saved the same amount of money?
b. How much will each person have saved at that time?
SOLUTION: a. Let w represent the number of weeks and t represent the total savings. So the two equations are:
Gregario: y = 35 + 10w Javier: y = 26 + 13w
Substitute 35 + 10w for t in the second equation to find the value of w.
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Using Addition and Subtraction
The solution is (−2, 2).
50. FINANCIAL LITERACY Gregorio and Javier each want to buy a bicycle. Gregorio has already saved $35 and plans to save $10 per week. Javier has
$26 and plans to save $13 per week.
a. In how many weeks will Gregorio and Javier have saved the same amount of money?
b. How much will each person have saved at that time?
SOLUTION: a. Let w represent the number of weeks and t represent the total savings. So the two equations are:
Gregario: y = 35 + 10w Javier: y = 26 + 13w
Substitute 35 + 10w for t in the second equation to find the value of w.
So, They will have the same amount of money saved in 3 weeks.
b. Substitute 3 for w in either equation.
Therefore, Gregorio and Javier will each have saved $65.
51. GEOMETRY A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.
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Using Addition and Subtraction
Therefore, Gregorio and Javier will each have saved $65.
51. GEOMETRY A parallelogram is a quadrilateral in which opposite sides are parallel. Determine whether ABCD is parallelogram. Explain your reasoning.
SOLUTION: Use the equations of the lines to find the slopes of the sides of the parallelogram.
lies on the vertical line described by x = –4, so its slope is undefined.
lies on the vertical line described by x = 3, so its slope is undefined.
lies on the line described by
, so it has a slope of .
lies on the line described by
, so it has a slope of .
Since each pair of opposite sides has the same slope, the opposite sides are parallel.
A quadrilateral with both pairs of opposite sides parallel is a parallelogram.
Therefore, ABCD is a parallelogram.
Solve each equation. Check your solution.
52. 6u = −48
SOLUTION: To check this answer, substitute −8 into the original equation. 6(−8) = −48, so the solution checks.
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53. 75 = −15p
SOLUTION: Page 57
6-3 Elimination Using Addition and Subtraction
To check this answer, substitute −8 into the original equation. 6(−8) = −48, so the solution checks.
53. 75 = −15p
SOLUTION: To check this answer, substitute −5 into the original equation. −15(−5) = 75, so the solution checks.
54. SOLUTION: To check this answer, substitute 12 into the original equation.
= 8, so the solution checks.
55. SOLUTION: To check this answer, substitute −20 into the original equation.
= 15, so the solution checks.
Simplify each expression. If not possible, write simplified.
56. 6q − 3 + 7q + 1
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SOLUTION: Page 58
6-3 Elimination
Addition
and−20
Subtraction
To check thisUsing
answer,
substitute
into the original equation.
= 15, so the solution checks.
Simplify each expression. If not possible, write simplified.
56. 6q − 3 + 7q + 1
SOLUTION: 2
2
57. 7w − 9w + 4w
SOLUTION: 58. 10(2 + r) + 3r
SOLUTION: 59. 5y − 7(y + 5)
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