Introductory Algebra
Name _________________________
Problem Set 7.2
Solutions to Every
Odd-Numbered Problem
Date _________________________
7.2 The Elimination Method
1.
5.
9.
11.
Adding the two equations:
3.
2x = 4
x=2
Substituting into the first equation:
2+y= 3
y =1
The solution is ( 2,1) .
Adding the two equations:
7.
!2 y = 10
y = !5
Substituting into the first equation:
x ! (!5) = 7
x+5=7
x=2
The solution is ( 2, !5 ) .
Adding the two equations:
0=0
The lines coincide (the system is dependent).
Multiplying the first equation by 2:
13.
6x ! 2y = 8
2 x + 2 y = 24
Adding the two equations:
8 x = 32
x=4
Substituting into the first equation:
3(4) ! y = 4
12 ! y = 4
! y = !8
y=8
The solution is ( 4, 8 ) .
Copyright © 2010 MathTV.com, Inc.
All rights reserved.
Videos at http://www.mathtv.com
Adding the two equations:
2 y = 14
y=7
Substituting into the first equation:
x + 7 = 10
x=3
The solution is ( 3, 7 ) .
Adding the two equations:
4 x = !4
x = !1
Substituting into the first equation:
!1 + y = !1
y=0
The solution is ( !1, 0 ) .
Multiplying the second equation by –3:
5 x ! 3y = !2
!30 x + 3y = !3
Adding the two equations:
!25 x = !5
1
x=
5
Substituting into the first equation:
! 1$
5 # & ' 3y = '2
" 5%
1 ' 3y = '2
'3y = '3
y =1
!1 $
The solution is # ,1& .
"5 %
This worksheet may be printed and used for educational purposes
only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
15.
19.
23.
Introductory Algebra
Name _________________________
Problem Set 7.2
Solutions to Every
Odd-Numbered Problem
Date _________________________
Multiplying the second equation by 4:
11x ! 4 y = 11
20 x + 4 y = 20
Adding the two equations:
31x = 31
x =1
Substituting into the second equation:
5(1) + y = 5
5+y=5
y=0
The solution is (1, 0 ) .
Multiplying the first equation by –2:
2 x + 16 y = 2
!2 x + 4 y = 13
Adding the two equations:
20 y = 15
3
y=
4
Substituting into the first equation:
" 3%
! x ! 8 $ ' = !1
# 4&
! x ! 6 = !1
!x = 5
x = !5
3%
"
The solution is $ !5, ' .
#
4&
Adding the two equations:
8 x = !24
x = !3
Substituting into the second equation:
2(!3) + y = !16
!6 + y = !16
y = !10
The solution is ( !3, !10 ) .
Copyright © 2010 MathTV.com, Inc.
All rights reserved.
Videos at http://www.mathtv.com
17.
21.
Multiplying the second equation by 3:
3x ! 5 y = 7
!3x + 3y = !3
Adding the two equations:
!2 y = 4
y = !2
Substituting into the second equation:
! x ! 2 = !1
!x = 1
x = !1
The solution is ( !1, !2 ) .
Multiplying the first equation by 2:
!6 x ! 2 y = 14
6 x + 7 y = 11
Adding the two equations:
5 y = 25
y=5
Substituting into the first equation:
!3x ! 5 = 7
!3x = 12
x = !4
The solution is ( !4, 5 ) .
This worksheet may be printed and used for educational purposes
only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
25.
Introductory Algebra
Name _________________________
Problem Set 7.2
Solutions to Every
Odd-Numbered Problem
Date _________________________
Multiplying the second equation by 3:
x + 3y = 9
6 x ! 3y = 12
Adding the two equations:
7 x = 21
x=3
Substituting into the first equation:
3 + 3y = 9
3y = 6
y=2
The solution is ( 3, 2 ) .
29.
27.
Multiplying the second equation by 2:
x ! 6y = 3
8 x + 6 y = 42
Adding the two equations:
9 x = 45
x=5
Substituting into the second equation:
4(5) + 3y = 21
20 + 3y = 21
3y = 1
1
y=
3
1
!
$
The solution is # 5, & .
" 3%
Multiplying the second equation by –3:
2x + 9y = 2
!15 x ! 9 y = 24
Adding the two equations:
!13x = 26
x = !2
Substituting into the first equation:
2(!2) + 9 y = 2
!4 + 9 y = 2
9y = 6
2
y=
3
2%
"
The solution is $ !2, ' .
#
3&
Copyright © 2010 MathTV.com, Inc.
All rights reserved.
Videos at http://www.mathtv.com
This worksheet may be printed and used for educational purposes
only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
31.
33.
35.
Introductory Algebra
Name _________________________
Problem Set 7.2
Solutions to Every
Odd-Numbered Problem
Date _________________________
To clear each equation of fractions, multiply the first equation by 12 and the second equation by 6:
1 $
1 %
!1
! 7$
"3
" 7%
12 # x + y& = 12 # &
6 $ x ! y' = 6 $ '
"3
%
"
%
#
&
# 3&
4
6
2
3
4 x + 3y = 14
9 x ! 2 y = 14
The system of equations is:
4 x + 3y = 14
9 x ! 2 y = 14
Multiplying the first equation by 2 and the second equation by 3:
8 x + 6 y = 28
27 x ! 6 y = 42
Adding the two equations:
35 x = 70
x=2
Substituting into 4 x + 3y = 14 :
4(2) + 3y = 14
8 + 3y = 14
3y = 6
y=2
The solution is ( 2, 2 ) .
Multiplying the first equation by –2:
!6 x ! 4 y = 2
6x + 4y = 0
Adding the two equations:
0=2
Since this statement is false, the two lines are parallel, so the system has no solution.
Multiplying the first equation by 2 and the second equation by 3:
22 x + 12 y = 34
15 x ! 12 y = 3
Adding the two equations:
37 x = 37
x =1
Substituting into the second equation:
5(1) ! 4 y = 1
5 ! 4y =1
!4 y = !4
y =1
The solution is (1,1) .
Copyright © 2010 MathTV.com, Inc.
All rights reserved.
Videos at http://www.mathtv.com
This worksheet may be printed and used for educational purposes
only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
37.
39.
41.
45.
Introductory Algebra
Name _________________________
Problem Set 7.2
Solutions to Every
Odd-Numbered Problem
Date _________________________
To clear each equation of fractions, multiply the first equation by 6 and the second equation by 6:
1 $
1 %
!1
! 1$
"
" 1%
6 # x + y& = 6 # &
6 $ ! x ! y' = 6 $ ! '
"2
%
"
%
#
&
# 6&
6
3
3
3x + y = 2
!6 x ! 2 y = !1
The system of equations is:
3x + y = 2
!6 x ! 2 y = !1
Multiplying the first equation by 2:
6x + 2y = 4
!6 x ! 2 y = !1
Adding the two equations:
0=3
Since this statement is false, the two lines are parallel, so the system has no solution.
Multiplying the second equation by 100 (to eliminate decimals):
x + y = 22
5 x + 10 y = 170
Multiplying the first equation by –5:
!5 x ! 5 y = !110
5 x + 10 y = 170
Adding the two equations:
5 y = 60
y = 12
Substituting into the first equation:
x + 12 = 22
x = 10
The solution is (10,12 ) .
Solving the equation:
43.
Solving the equation:
x + ( 2 x ! 1) = 2
2 ( 3y ! 1) ! 3y = 4
x + 2x !1 = 2
6 y ! 2 ! 3y = 4
3x ! 1 = 2
3y ! 2 = 4
3x = 3
3y = 6
x =1
y=2
Solving the equation:
4 x + 2 ( !2 x + 4 ) = 8
4x ! 4x + 8 = 8
8=8
Since this statement is true, the solution is all real numbers.
Copyright © 2010 MathTV.com, Inc.
All rights reserved.
Videos at http://www.mathtv.com
This worksheet may be printed and used for educational purposes
only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
47.
49.
51.
53.
55.
Introductory Algebra
Name _________________________
Problem Set 7.2
Solutions to Every
Odd-Numbered Problem
Date _________________________
Solving for x:
x ! 3y = !1
x = 3y ! 1
Solving for y: y = 2 (1) ! 1 = 2 ! 1 = 1
Solving for x: x = 3( 2 ) ! 1 = 6 ! 1 = 5
Substituting x = 13: y = 1.5 (13) + 15 = 19.5 + 15 = 34.5
Substituting x = 12: y = 0.75 (12 ) + 24.95 = 9 + 24.95 = 33.95
Copyright © 2010 MathTV.com, Inc.
All rights reserved.
Videos at http://www.mathtv.com
This worksheet may be printed and used for educational purposes
only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.