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How is the best way to do the systems of equations?

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There is not a hands-down, "best way" - it depends on the job you are trying to do, just like there is no "best hammer" in a handyman's toolbox. 

The two most common ways are elimination and substitution (I'll spare describing matrices). 

In Elimination, you combine the two equations with the purpose of eliminating one of the variables. In this case, you may need to multiply one or more equations by a constant to get the coefficients of the term you want to eliminate equal. For example, if we had the system

2x – y = 9
3x + 4y = –14 

we could multiply the first equation by 4, to get the same coefficient in front of y:

(4)2x - (4)y = (4)(9) 

8x - 4y  = 36
3x + 4y = -14 

We can now add the two together to eliminate y, and we're left with one equation in terms of x:

11x = 22

x = 2

We put this back into EITHER equation and solve for y. 


Substitution, we solve one for one of the variables and replace that variable in the other equation with the new value. Take the same system:

2x – y = 9
3x + 4y = –14

solve the first one for y:   y = 2x - 9

and now replace it in the second for y:

3x + 4(2x - 9) = -14

3x + 8x -36 = -14

11x = 22

x = 2

Now plug this into either equation and solve for y. (Since you already have y = from the first equation, that sounds like a good place to put it). 

It really depends on preference, but after getting used to both methods, it will be easier to see which one is the better tool for a specific problem. 

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