Someone please explain in the most simple way possible please. (edit: i am using elimination)
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In: 1
You know the equation of a line, right?
Well, what if you had 2 lines and you wanted to know the x & y coordinate where they meet (if at all)? You could graph it, but you can’t tell if they cross at x=5 or x=5.02 (unless using a computer).
Substitution & elimination are ways to solve it algebraically.
[**I wrote a how-to**](https://docs.google.com/document/d/1OevWqMCtjjAhEy20W1YTwmyyyQIW0WzZ6LvhcrGGdK0/export?format=pdf).
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To use the example from my write-up:
4x – 3y = 5
3x + 2y = 8
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**Substitution**
If we isolate x in the top equation we get:
x = (5 + 3y)/4
Since their x & y values have to be identical, we can plug (substitute) this “value” of x into the second equation, leaving only y variables which we can isolate:
3( (5 + 3y)/4) ) + 2y = 8
(15 + 9y)/4 + 2y = 8
(15 + 9y)/4 = 8 – 2y
15 + 9y = 32 – 8y
17y = 17
**y = 1**
We can now plug this into either 2 equations (including the isolated x equation):
x = (5 + 3y)/4
x = (5 + 3)/4
x = 8/4
**x = 2**
Thus **(2,1)** is where they cross.
You could have isolated y first, it doesn’t matter.
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**Elimination**
Just like substitution, we want to get rid of 1 variable so we can isolate the other. However, this way is different in that you combine the equations, turning them into equation equations if need be.
4x – 3y = 5
3x + 2y = 8
We can multiple the coefficients of a selected variable to obtain a GCF/LCM, in this case multiple the 1st & 2nd equations by 3 & 4 respectively for x :
12x – 9y = 15
12x + 8y = 32
We can now subtract the equations:
-17y = -17
**y = 1**
We can now substitute that in or just re-do the elimination for y, I’ll do the latter by multiplying by 8 & 9 respectively:
96x – 72y = 120
108x + 72y = 288
We can now add the equations
204x = 408
**x = 2**
Thus **(2,1)** is where they cross.
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When you have 3+ equations and/or 3+ variables, it’s recommend to do Gauss-Jordan elimination visualization, as it saves a lot of space (you just write the coefficients & constants, not the variables, sort of like synthetic division). I can’t give you an example right now as it’s a pain to type. However, here is TOCT’s video on it: https://youtu.be/eYSASx8_nyg
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