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This module teaches elimination as a means to solve simultaneous equations.

Here is the algorithm for elimination.

  • Multiply one equation (or in some cases both) by some number, so that the two equations have the same coefficient for one of the variables.
  • Add or subtract the two equations to make that variable go away.
  • Solve the resulting equation, which now has only one variable.
  • Finally, plug back in to find the other variable.

Solving simultaneous equations by elimination

3x + 4y = 1 size 12{3x+4y=1} {}

2x y = 8 size 12{2x - y=8} {}

  • 1

    The first question is: how do we get one of these variables to have the same coefficient in both equations ? To get the x size 12{x} {} coefficients to be the same, we would have to multiply the top equation by 2 and the bottom by 3. It is much easier with y size 12{y} {} ; if we simply multiply the bottom equation by 4, then the two y size 12{y} {} values will both be multiplied by 4.
    • 3x + 4y = 1 size 12{3x+4y=1} {}
    • 8x 4y = 32 size 12{8x - 4y="32"} {}
  • 2

    Now we either add or subtract the two equations. In this case, we have 4y size 12{4y} {} on top, and 4y size 12{ - 4y} {} on the bottom; so if we add them, they will cancel out. (If the bottom had a + 4y size 12{+4y} {} we would have to subtract the two equations to get the " y size 12{y} {} "s to cancel.)
    • 11 x + 0y = 33 size 12{"11"x+0y="33"} {}
  • 3-4

    Once again, we are left with only one variable. We can solve this equation to find that x = 3 size 12{x=3} {} and then plug back in to either of the original equations to find y = 2 size 12{y= - 2} {} as before.

Why does elimination work?

As you know, you are always allowed to do the same thing to both sides of an equation. If an equation is true, it will still be true if you add 4 to both sides, multiply both sides by 6, or take the square root of both sides.

Now—consider, in the second step above, what we did to the equation 3x + 4y = 1 size 12{3x+4y=1} {} . We added something to both sides of this equation. What did we add? On the left, we added 8x 4y size 12{8x - 4y} {} ; on the right, we added 32. It seems that we have done something different to the two sides.

However, the second equation gives us a guarantee that these two quantities, 8x 4y size 12{8x - 4y} {} and 32, are in fact the same as each other . So by adding 8x 4y size 12{8x - 4y} {} to the left, and 32 to the right, we really have done exactly the same thing to both sides of the equation 3x + 4y = 1 size 12{3x+4y=1} {} .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
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is it a question of log
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Commplementary angles
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
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Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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Smarajit Reply
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Source:  OpenStax, Simultaneous equations. OpenStax CNX. Mar 23, 2011 Download for free at http://cnx.org/content/col11287/1.1
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