6-3 Elimination Using Addition and Subtraction
Use elimination to solve each system of equations.
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.
The solution is (−3, −4).
Check the solution in each equation.
6y = 78
15. 9x +Manual
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3x − 6y = −30
SOLUTION: Page 1
6-3 Elimination Using Addition and Subtraction
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.
The solution is (4, 7).
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.
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6-3 Elimination Using Addition and Subtraction
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.
The solution is (1, 2.5).
Check the solution in each equation.
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.
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Because
and y have
opposite
coefficients, add the equations.
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6-3 Elimination Using Addition and Subtraction
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.
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.
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6-3 Elimination
Using Addition and Subtraction
The two numbers are 2 and 9.
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).
27. 6x − 7y = −26
6x + 5y = 10
SOLUTION: Because 6x and 6x have the same coefficients, subtract the equations.
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Now, substitute 3 for y in either equation to find the value of x.
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6-3 Elimination
Using
Addition and Subtraction
The solution is
(4, −1).
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
.
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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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.
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
4, or 5.
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number
of -pennies
1, 2, 3,
d. VERBAL Does the result in the table match the results in part b? Explain.
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b. ANALYTICAL Write and solve a system of equations to find the number of paper clips and pennies used for
15 points. Using Addition and Subtraction
6-3 Elimination
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.
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.
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).
Describe
39. WRITING IN MATH eSolutions
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equations.
SOLUTION: when it would be most beneficial to use elimination to solve a system of
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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
6-3 Elimination
Using Addition
andthe
Subtraction
and b have opposite
coefficients,
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).
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.
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