Elimination

= __ELIMINATION __ =

Algorithms: 1. Arrange the variables so that the x, the y and the c are aligned together. x + 4y = 16  y = -7x - 13   x + 4y = 16   7x + y = -13   2. After you have aligned the variables, you must match a certain variable in the first equation to the second equation. Distribute if necessary (for this situation, we have to distribute a number that would match the "y" from the second equation to the "y" in the first equation, our chosen variable.)   4 ﻿( 7x + y = -13) ﻿   //28x + 4y = -52// 3. You may now solve the equations by using either addition or subtraction (in this case, we are going to use subtraction to eliminate "y") and then solve for "x".

x + 4y = 16  - 28x + 4y = -52   -27x = 68   -27x/-27 = 68/-27   //x = -2.518//   4. You may now substitute the "x" into any equations mentioned above to get the remaining variable, "y". If you want life to be easier for you, use the simpler one, such as the first one. x + 4y = 16 -2.158 + 4y = 16   4y = 16 + 2.518   4y = 18.518   4y/4 = 18.518/4   y = 4.6295 5. Now that "y" is solved for, you may now substitute both "x" and "y" in any equation you want to use and solve it. You may still use the same equation you used for #4.

x + 4y = 16 -2.158 + 4(4.6295) = 16   -2.518 + 18.518 = 16   16 = 16   There you have it! I hope it helped you understand elimination ^_^  If you still haven't figured it out, here are some more examples:   4x + 5y= -9   8x-y=-7   First, we must solve for either variable x or variable y.   So lets solve for variable y first. 1). The first thing we must always do, is move all the x's on one side, and all the y's on one side, and the other number without a variable on one side, but in this case, __//we dont need to do that because all the x's, y's and other numbers are lined up//.__ The equation will still look like this : 4x + 5y = -9  8x-y=-7  2). So the next step is to DISTRIBUTE. Now we will multiply the first equation with -8, and the 2nd equation with 4. So it will look like this: -8 (4x + 5y = -9) 4 (8x - y = -7)

*Note that in this situation, we must multiply both of the equations with the same number from each equation, so that we can get the same number.

We should change one of these numbers into a negative number as well, so that when we add them up, we will get ZERO.

SO now the equation will look like this:

-32x - 40y = 72 32x - 4y = -28

3). Next step: ELIMINATE X

Since x in both equations are now the same number, we should then eliminate the x. To eliminate the x, we use ADDITION , because when we add -32x and 32 x, we will get ZERO.

//__*The only reason we do not subtract, is because if we subtract 32x, from -32x, we would get 64. However, we want to get 0.__//

Now the equation will look like this:

-44y=44

4). Then we divide both sides by -44, so it will now look like this:

-44y/-44 = 44/-44

5). Get the final result of y. Now -44 immediately cancels out -44y, and 44/-44 is equal to negative 1.

SO THE ANSWER FOR y= -1 !

Now that we have y, we must solve for the variable x.

6). We take any of the two starting equations, and SUBSTITUTE our y, into the equation.

So lets take this equation: 4x + 5y = -9

When we substitute it, it will now be like this: 4x + 5(-1) = -9

7). We then multiply 5 by -1. It will now look like this :

4x -5 = -9

8). Again, we move the x on one side, and the other numbers on the other side:

4x= -9 +5 4x= -4

Divide both sides by 4:

4x/4 = -4/4 x= -1

9). Now finalize your answers = x=-1 = = y=-1 ﻿Now that we have our answers, we need to check if our answers are correct. In order to do that we need to substitue one of our answers into any of our equations.  =  ** So let's chose one equation: 4x + 5y -9 ** 1). When we substitue it will look like this: 4(-1) + 5y = -9 -4 + 5y = -9 5y = -9 + 4 5y= -5 5y/5= -5/5 y= -1 **Now because we substituted x into our equation, we know that our answer is correct if the answer from the equation above matches our original answer, which is y = -1.**

= NOW TRY THIS ON YOUR OWN! :-) = 2x + 5y =-9  8x - y =-7 To check your work, click here.