The objective of elimination is to eliminate one of the variables in either equation by adjusting the other equation and adding them. In the example above, we might multiply the top equation by (-d/a) to get:
-(d/a)*ax + -(d/a)*by = -(d/a)*c
dx + ey = f
-dx + -dby/a = -dc/a
dx + ey = f
When I add eqation 1 to equation 2, I eliminate x from equation 2:
-dx + -dby/a + dx + ey= -dc/a + f
-dby/a + ey = -dc/a + f or (-db/a + e)y = -dc/a + f
Once you have solved for y, you can use that solution to solve for x.
Consider a real example:
3x + y = 11
2x - 2y = 10
When I look at this equation, the first thing I consider is eliminating y in the first equation, it looks like the easiest thing to do. So I multiply the first equation by 2:
6x + 2y = 22
2x - 2y = 10
Then I add equation 1 and 2:
6x + 2y + 2x = 2y = 22 + 10
which reduces to 8x = 32 or
Now that I have this result, I substitute x into equation 1 and I get:
which reduces to:
adding 8 to both sides leaves me with:
dividing both side by -2 I get
Of course you should check your answer:
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