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5 – 3: Solving a System of Two Equations by Elimination
Selected Worked Homework Problems
Solve each system by the elimination method. List your answers as an ordered pair.
1.
�
Equation AEquation B
3x + 5y = 2−3x + y = −14⎧ ⎨ ⎩
!
�
If you add the left sides of Equation A and Equation B togetherand add the right sides of Equation A and Equation B togetherthe x terms add to zero and you will have eliminated the x terms. You now have a new equation with only the y variable.
Add Equation A and Equation B to eliminate the x terms
3x + 5y = 2−3x + 1y = −146y = −12Solve for yy = −2
Plug y = −2 into either equation A or Band solve for x
Equation A 3x + 5(−2) = 23x −10 = 23x = 12x = 4Answer: (4,−2)
check:
Equation A Equation B3x + 5y = 2 − 3x + 1y = −143(4) + 5(−2) = 2 − 3(4) + 1(−2) = −1412 −10 = 2 −12 − 2 = −14
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
2.
�
Equation AEquation B
3x − 2y = −11−5x + 2y = 21⎧ ⎨ ⎩
�
If you add the left sides of Equation A and Equation B togetherand add the right sides of Equation A and Equation B togetherthe y terms add to zero and you will have eliminated the y terms. You now have a new equation with only the x variable.
Add Equation A and Equation B to eliminate the y terms
3x − 2y = −11−5x + 2y = 21−2x = 10Solve for yx = −5
Plug x = −5 into either equation A or Band solve for y
Equation A 3x − 2y = −113(−5) − 2y = −11−15 − 2y = −11−2y = 4y = −2Answer: (−5,−2)
check:
Equation A Equation B3x − 2y = −11 − 5x + 2y = 213(−5) − 2(−2) = −11 − 5(−5) + 2(−2) = 21−15 + 4 = −11 25 − 4 = 21
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
3.
�
Equation AEquation B
2x − y = −3−2x + 3y = −9⎧ ⎨ ⎩
�
If you add the left sides of Equation A and Equation B togetherand add the right sides of Equation A and Equation B togetherthe x terms add to zero and you will have eliminated the x terms. You now have a new equation with only the y variable.
Add Equation A and Equation B to eliminate the x terms
2x − y = −3−2x + 3y = −92y = −12Solve for yy = −6
Plug y = −6 into either equation A or Band solve for x
Equation A 2x − (y) = −32x − (−6) = −32x + 6 = −32x = −9
x =−92
Answer: -92
,−6⎛ ⎝
⎞ ⎠
check:
Equation A Equation B2x − y = −3 − 2x + 3y = −9
2 −92
⎛ ⎝
⎞ ⎠ − (−6) = −3 − 2 −9
2⎛ ⎝
⎞ ⎠ + 3(−6) = −9
−9 + 6 = −3 9 −18 = −9
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
4.
�
Equation AEquation B
4x − 5y = −3−4x + 5y = 1⎧ ⎨ ⎩
If you add the left sides of Equation A and Equation B togetherand add the right sides of Equation A and Equation B togetherthe y terms add to zero and you will have eliminated the y terms. You now have a new equation with only the x variable.
Add Equation A and Equation B to eliminate the y terms
4x − 5y = −3−4x + 5y = 10 = −2
STOP: Both the x and y terms canceled outand the remaining equation 0 = −2 is false
The lines are parallel,they have no common points.
Answer: No Solution
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
5.
�
Equation AEquation B
2x − y = −4−2x + y = 4⎧ ⎨ ⎩
!
If you add the left sides of Equation A and Equation B togetherand add the right sides of Equation A and Equation B togetherthe y terms add to zero and you will have eliminated the y terms. You now have a new equation with only the x variable.
Add Equation A and Equation B to eliminate the y terms
2x − y = −4−2x + y = 40 = −2
STOP: Both the x and y terms canceled outand the remaining equation 0 = 0 is true
Both equations describe the same lineany point on 2x − y = −4 would also be on − 2x + y = 4
Answer: All Points on 2x − y = −4orAnswer: All Points on − 2x + y = 4 either one of the above is correct
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
6.
�
Equation AEquation B
− x − 3y = −6−2x + 3y = 0⎧ ⎨ ⎩
�
If you add the left sides of Equation A and Equation B togetherand add the right sides of Equation A and Equation B togetherthe y terms add to zero and you will have eliminated the y terms. You now have a new equation with only the x variable.
Add Equation A and Equation B to eliminate the y terms
− x − 3y = −6−2x + 3y = 0−3x = −6Solve for yx = 2
Plug x = 2 into either equation A or Band solve for y
Equation A −x − 3y = −6−(2) − 3y = −6−2 − 3y = −6−3y = −4
y =43
Answer: 2, 43
⎛ ⎝
⎞ ⎠
check:
Equation A Equation B−x − 3y = −6 − 2x + 3y = 0
−(2) − 3 43
⎛ ⎝
⎞ ⎠ = −6 − 2(2) + 3 4
3⎛ ⎝
⎞ ⎠ ) = 0
−2 − 4 = −6 − 4 + 4 = 0
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
7.
�
Equation AEquation B
2x + 5y = 93x + y = 7⎧ ⎨ ⎩
! 8. Equation AEquation B
2x − 3y = 5−4x + 2y = 2
⎧⎨⎩
�
Multiply Equation B by − 5 so thatEquation A has 5x andEquation B has − 5x
−52x + 5y = 93x + y = 7⎧ ⎨ ⎩
2x + 5y = 9−15x − 5y = −35 −13x = −26 x = 2
Plug x = 2 into either equation A or Band solve for y
Equation A 2(2) + 5y = 94 + 5y = 95y = 5y = 1Answer: (2,1 )
! !
�
Multiply Equation A by 2 so thatEquation A has 4x andEquation B has − 4x
2 2x − 3y = 5−4x + 2y = 2⎧ ⎨ ⎩
4x − 6y = 10−4x + 2y = 2 − 4y =12 y = −3
Plug y = −3 into either equation A or Band solve for x
Equation A 2(x) − 3 −3( ) = 52x + 9 = 52x = −4x = −2Answer: (− 2,−3 )
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
9.
�
Equation AEquation B
3x + y = −7x + 2y = 6
⎧ ⎨ ⎩
! 10.
�
−4x + 5y = −168x + y = −1⎧ ⎨ ⎩
�
Multiply Equation B by − 3 so thatEquation A has 3x andEquation B has − 3x
−33x + y = −7x + 2y = 6
⎧ ⎨ ⎩
3x + y = −7−3x − 6y = −18 − 5y = −25 y = 5
Plug y = 5 into either equation A or Band solve for x
Equation A 3(x) + y( ) = −73(x) + 5( ) = −73x + 5 = −73x = −12x = −4Answer: (− 4,5 )
!
�
Multiply Equation A by 2 so thatEquation A has − 8x andEquation B has 8x
2 −4x + 5y = −168x + y = −1⎧ ⎨ ⎩
−8x + 10y = −32 8x + y = −1 11y = −33 y = −3
Plug y = −3 into either equation A or Band solve for x.
Equation A−4(x) + 5(−3) = −16 −4x −15 = −16−4x = −1
x =14
Answer: 14
,−3⎛ ⎝
⎞ ⎠
NOTE: You could have multiplied! NOTE: You could have multiplied Equation A by –2 instead of! Equation B by –2 instead of multiplying Equation B by –3! multiplying Equation B by –5 This would eliminate the y terms ! This would eliminate the y terms and you would then solve for x.! and you would then solve for x.
You get the same answer in either case.
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
14.
�
Equation AEquation B
2x + 3y = −85x + 4y = −34⎧ ⎨ ⎩
�
You must mutiply both rows by different numbers to eliminate a variable
Multiply Equation A by 5Multiply Equation B by − 2to eliminate the x terms
5−2
2x + 3y = −85x + 4y = −34⎧ ⎨ ⎩
10x + 15y = −40−10x − 8y = 68 Now add the two equations 7y = 28 Solve for y y = 4
Plug y = 4 into either equation A or Band solve for x
Equation A 2x + 3(y) = −8 2x + 3(4) = −82x + 12 = −82x = −20x = −10Answer: −10,4( )
NOTE: You could have multiplied Equation A by –2! instead of multiplying Equation B by –3! ! This would eliminate the y terms ! and you would then solve for x.!
You get the same answer in either case.
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
15.
�
Equation AEquation B
3x − 5y = 114x + 3y = 5⎧ ⎨ ⎩
!
You must mutiply both rows by different numbers to eliminate a variable
Multiply Equation A by 3Multiply Equation B by 5to eliminate the y terms
35
3x − 5y = 114x + 3y = 5
⎧⎨⎩
9x −15y = 3320x +15y = 25 Now add the two equations 29x = 58 Solve for y x = 2
Plug x = 2 into either equation A or Band solve for y
Equation A 3(x)− 5(y) = 11 3(2)− 5(y) = 116 − 5y = 11−5y = 5y = −1Answer: 2,−1( )
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
21.
�
Equation A
Equation B
x3−
3y4
=−12
x6
+y8
=34
⎧
⎨ ⎪
⎩ ⎪
!
�
Multiply Equation A by 12Multiply Equation B by 24to eliminate the fractions
12
24
x3−
3y4
=−12
x6
+y8
=34
⎧
⎨ ⎪
⎩ ⎪
to get 2 new equations
Equation CEquation D
4x − 9y = −64x + 3y = 18⎧ ⎨ ⎩
which you now can solve
�
Multiply Equation A by −1to eliminate the x terms
−1 4x − 9y = −64x + 3y = 18⎧ ⎨ ⎩
−4x + 9y = 64x + 3y = 18 Now add the two equations 12y = 24 Solve for y y = 2
Plug y = 2 into either equation A or Band solve for x
Equation A 4(x) − 9(y) = −6 4(x) − 9(2) = −6 4x −18 = −64x = 12x = 3
Answer: 3,2( )Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel
22. Equation A
Equation B
x2− y
4= 3
4x2+ y
6= 7
6
⎧
⎨⎪⎪
⎩⎪⎪
�
Multiply Equation A by 4Multiply Equation B by 6to eliminate the fractions
4
6
x2−y4
=34
x2
+y6
=76
⎧
⎨ ⎪
⎩ ⎪
to get 2 new equations
Equation CEquation D
2x − y = 33x + y = 7⎧ ⎨ ⎩
which you now can solve
Add Equation C and Equation D to eliminatethe y terms and solve for x
Equation CEquation D
2x − y = 33x + y = 7
⎧⎨⎩
5x = 10 x = 2
Plug x = 2 into either equation C or Dand solve for x
Equation C 2(x)− (y) = 3 2(2)− (y) = 3 4 − y = 3−y = −1y = 1
Answer: 2,1( )
Math 100 ! Section 5 – 3 HW WKD ! © 2016 Eitel