Intermediate Algebra Name Problem Set 4.1 Odd-Numbered … · 2010-11-02 · 8x+6y=28 27x!6y=42...
Transcript of Intermediate Algebra Name Problem Set 4.1 Odd-Numbered … · 2010-11-02 · 8x+6y=28 27x!6y=42...
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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.
4.1 Systems of Linear Equations in Two Variables 1. The intersection point is (4,3). 3. The intersection point is (–5,–6).
5. The intersection point is (4,2).
7. The lines are parallel. There is no solution to the system.
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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.
9. Solving the two equations:
3x + y = 5
3x ! y = 3
Adding yields:
6x = 8
x =4
3
Substituting into the first equation:
34
3
!"#
$%&+ y = 5
4 + y = 5
y = 1
The solution is 4
3,1
!"#
$%&
.
11. Multiply the first equation by 3:
3x + 6y = 0
2x ! 6y = 5
Adding yields:
5x = 5
x = 1
The solution is 1,!1
2
"#$
%&'
.
13. Multiply the first equation by –2:
!4x +10y = !32
4x ! 3y = 11
Adding yields:
7y = !21
y = !3
The solution is 1
2,!3"
#$%&'
.
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
Copyright © 2010 MathTV.com, Inc. All rights reserved. Videos at http://www.mathtv.com
This worksheet may be printed and used for educational purposes only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
15. Multiply the first equation by 3 and the second equation by –2:
18x + 9y = !3
!18x !10y = !2
Adding yields:
!y = !5
y = 5
The solution is !8
3,5
"#$
%&'
.
17. Multiply the first equation by 2 and the second equation by 3:
8x + 6y = 28
27x ! 6y = 42
Adding yields:
35x = 70
x = 2
The solution is (2,2). 19. Multiply the first equation by 2:
4x !10y = 6
!4x +10y = 3
Adding yields 0 = 9, which is false. There is no solution (! ). 21. To clear each equation of fractions, multiply the first equation by 6 and the second equation by 20:
3x + 2y = 78
8x + 5y = 200
Multiply the first equation by 5 and the second equation by –2:
15x +10y = 390
!16x !10y = !400
Adding yields:
!x = !10
x = 10
The solution is (10,24).
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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.
23. To clear each equation of fractions, multiply the first equation by 15 and the second equation by 6:
10x + 6y = !60
2x ! 3y = !2
Multiply the second equation by 2:
10x + 6y = !60
4x ! 6y = !4
Adding yields:
14x = !64
x = !32
7
Substituting into the second equation:
2 !32
7
"#$
%&'! 3y = !2
!64
7! 3y = !2
!3y =50
7
y = !50
21
The solution is !32
7,!50
21
"#$
%&'
.
25. Substituting into the first equation: 27. Substituting into the first equation:
7 2y + 9( ) ! y = 24
14y + 63! y = 24
13y = !39
y = !3
6x ! !3
4x !1"
#$%&'= 10
6x +3
4x +1 = 10
27
4x = 9
27x = 36
x =4
3
The solution is (3,–3). The solution is 4
3,!2"
#$%&'
.
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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.
29. Substituting into the first equation:
4x ! 4 = 3x ! 2
x ! 4 = !2
x = 2
The solution is (2,4). 31. Solving the first equation for y yields y = 2x ! 5 . Substituting into the second equation:
4x ! 2 2x ! 5( ) = 10
4x ! 4x +10 = 10
10 = 10
Since this statement is true, the two lines coincide. The solution is (x, y) | 2x ! y = 5{ } . 33. Substituting into the first equation:
1
3
3
2y
!"#
$%&'1
2y = 0
1
2y '
1
2y = 0
0 = 0
Since this statement is true, the two lines coincide. The solution is (x, y) | x =3
2y
!"#
$%&
.
35. Multiply the first equation by 2 and the second equation by 7:
8x !14y = 6
35x +14y = !21
Adding yields:
43x = !15
x = !15
43
Substituting into the original second equation:
5 !15
43
"#$
%&'+ 2y = !3
!75
43+ 2y = !3
2y = !54
43
y = !27
43
The solution is !15
43,!27
43
"#$
%&'
.
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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. Multiply the first equation by 3 and the second equation by 8:
27x ! 24y = 12
16x + 24y = 48
Adding yields:
43x = 60
x =60
43
Substituting into the original second equation:
260
43
!"#
$%&+ 3y = 6
120
43+ 3y = 6
3y =138
43
y =46
43
The solution is 60
43,46
43
!"#
$%&
.
39. Multiply the first equation by 2 and the second equation by 5:
6x !10y = 4
35x +10y = 5
Adding yields:
41x = 9
x =9
41
Substituting into the original second equation:
79
41
!"#
$%&+ 2y = 1
63
41+ 2y = 1
2y = '22
41
y = '11
41
The solution is 9
41,!11
41
"#$
%&'
.
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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.
41. Multiply the second equation by 3:
x ! 3y = 7
6x + 3y = !18
Adding yields:
7x = !11
x = !11
7
Substituting into the original second equation:
2 !11
7
"#$
%&'+ y = !6
!22
7+ y = !6
y = !20
7
The solution is !11
7,!20
7
"#$
%&'
.
43. Substituting into the first equation:
!1
3x + 2 =
1
2x +
1
3
6 !1
3x + 2
"#$
%&'= 6
1
2x +
1
3
"#$
%&'
!2x +12 = 3x + 2!5x = !10x = 2
Substituting into the first equation: y =1
22( ) +
1
3= 1+
1
3=4
3. The solution is 2,
4
3
!"#
$%&
.
45. Substituting into the first equation:
32
3y ! 4"
#$%&'! 4y = 12
2y !12 ! 4y = 12!2y !12 = 12
!2y = 24y = !12
Substituting into the second equation: x =2
3!12( ) ! 4 = !8 ! 4 = !12 . The solution is (–12,–12).
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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. Multiply the first equation by 2:
8x ! 6y = !14
!8x + 6y = !11
0 = !25
Since this statement is false, there is no solution (! ). 49. Multiply the first equation by –20:
!60y ! 20z = !340
5y + 20z = 65
Adding yields:
!55y = !275
y = 5
Substituting into the first equation:
3 5( ) + z = 17
15 + z = 17
z = 2
The solution is y = 5, z = 2. 51. Substitute into the first equation:
3
4x !
1
3
1
4x
"#$
%&'= 1
3
4x !
1
12x = 1
2
3x = 1
x =3
2
Substituting into the second equation: y =1
4
3
2
!"#
$%&=3
8. The solution is
3
2,3
8
!"#
$%&
.
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
Copyright © 2010 MathTV.com, Inc. All rights reserved. Videos at http://www.mathtv.com
This worksheet may be printed and used for educational purposes only. It cannot be used for commercial purposes without the written
consent of MathTV.com, Inc. Created by Ross Rueger.
53. To clear each equation of fractions, multiply the first equation by 12 and the second equation by 12:
3x ! 6y = 4
4x ! 3y = !8
Multiply the second equation by –2:
3x ! 6y = 4
!8x + 6y = 16
Adding yields:
!5x = 20
x = !4
Substituting into the first equation:
3 !4( ) ! 6y = 4
!12 ! 6y = 4
!6y = 16
y = !8
3
The solution is !4,!8
3
"#$
%&'
.
55. a. Simplifying: 3x ! 4 y( ) ! 3 x ! y( ) = 3x ! 4 y ! 3x + 3y = ! y b. Substituting x = 0:
3 0( ) ! 4y = 8
!4y = 8
y = !2
c. From part b, the y-intercept is (0,–2). d. Graphing the line:
Intermediate Algebra Problem Set 4.1 Solutions to Every Odd-Numbered Problem
Name _________________________ Date _________________________
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.
e. Multiply the second equation by –3:
3x ! 4 y = 8!3x + 3y = !6
Adding yields:
! y = 2y = !2
Substituting into the first equation:
3x ! 4 !2( ) = 83x + 8 = 8
3x = 0
x = 0
The lines intersect at the point (0,–2). 57. Multiply the second equation by 100:
x + y = 10000
6x + 5y = 56000
Multiply the first equation by –5:
!5x ! 5y = !50000
6x + 5y = 56000
Adding yields x = 6000. The solution is (6000,4000). 59. Substituting x = 4 ! y into the second equation:
4 ! y( ) ! 2y = 44 ! 3y = 4
!3y = 0
y = 0
The solution is (4,0). 61. Simplifying: 2 ! 2 6( ) = 2 !12 = !10 63. Simplifying: x + 3y( ) !1 x ! 2z( ) = x + 3y ! x + 2z = 3y + 2z 65. Solving the equation: 67. Solving the equation:
!9y = !9
y = 1
3 1( ) + 2z = 93+ 2z = 9
2z = 6
z = 3
69. Applying the distributive property: 2 5x ! z( ) = 10x ! 2z 71. Applying the distributive property: 3 3x + y ! 2z( ) = 9x + 3y ! 6z