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Elimination

Module by: Kenny M. Felder. E-mail the author

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

Here is the algorithm for elimination.

  1. 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.
  2. Add or subtract the two equations to make that variable go away.
  3. Solve the resulting equation, which now has only one variable.
  4. Finally, plug back in to find the other variable.

Example 1: Solving Simultaneous Equations by Elimination

3x+4y=13x+4y=1 size 12{3x+4y=1} {}

2xy=82xy=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 xx 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 yy size 12{y} {}; if we simply multiply the bottom equation by 4, then the two yy size 12{y} {} values will both be multiplied by 4.
    • 3x+4y=13x+4y=1 size 12{3x+4y=1} {}
    • 8x4y=328x4y=32 size 12{8x - 4y="32"} {}
  • 2: Now we either add or subtract the two equations. In this case, we have 4y4y size 12{4y} {} on top, and 4y4y size 12{ - 4y} {} on the bottom; so if we add them, they will cancel out. (If the bottom had aa +4y+4y size 12{+4y} {} we would have to subtract the two equations to get the " yy size 12{y} {}"s to cancel.)
    • 11x+0y=3311x+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=3x=3 size 12{x=3} {} and then plug back in to either of the original equations to find y=2y=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=13x+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 8x4y8x4y 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, 8x4y8x4y size 12{8x - 4y} {} and 32, are in fact the same as each other. So by adding 8x4y8x4y 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=13x+4y=1 size 12{3x+4y=1} {}.

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