5 – 3: solving a system of two equations by elimination ...wserver.flc.losrios.edu/~eitel/all...

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⎧ ⎨ ⎩

!

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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.

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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


3.

�


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


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


4.

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4x − 5y = −3−4x + 5y = 1⎧ ⎨ ⎩



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


5.

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2x − y = −4−2x + y = 4⎧ ⎨ ⎩

!



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


6.

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− x − 3y = −6−2x + 3y = 0⎧ ⎨ ⎩

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− 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


7.

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2x + 5y = 93x + y = 7⎧ ⎨ ⎩

! 8. Equation AEquation B

2x − 3y = 5−4x + 2y = 2

⎧⎨⎩

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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


Equation A 2(2) + 5y = 94 + 5y = 95y = 5y = 1Answer: (2,1 )

! !

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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


Equation A 2(x) − 3 −3( ) = 52x + 9 = 52x = −4x = −2Answer: (− 2,−3 )


9.

�


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 )

!

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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.


14.

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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


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.


15.

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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


Equation A 3(x)− 5(y) = 11 3(2)− 5(y) = 116 − 5y = 11−5y = 5y = −1Answer: 2,−1( )


21.

�

Equation A

Equation B

x3−

3y4

=−12

x6

+y8

=34

⎧

⎨ ⎪

⎩ ⎪

!

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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

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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


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


2x − y = 33x + y = 7⎧ ⎨ ⎩

which you now can solve

Add Equation C and Equation D to eliminatethe y terms and solve for x


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( )


5 – 3: solving a system of two equations by elimination ...wserver.flc.losrios.edu/~eitel/all...

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