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If I am asked to solve a systems of equation, how would I know which method (substitution, or elimination) to use? What set of conditions should I be looking for, or is it that either method should in theory work?
Then say that I am given the following system of equation, but neither substituion nor elimination produces the correct result (based on what I have did thus far)
$xy = 4$
$2x - y - 7 = 0$
How would I go on about solving this type of one?
Solve for one variable using the second equation. Let's solve for $y$: $$2x - y - 7 = 0 \iff y = 2x - 7$$
Now plug $y = 2x - 7$ into $xy = 4$ to get a quadratic in $x$:
$$xy = 4 \iff x(2x - 7) = 4 \iff 2x^2 - 7x - 4 = 0 \iff (2x + 1)(x - 4) = 0$$
If the factoring isn't immediately apparent, you can use the quadratic formula to obtain two solutions: $x = -1/2$ or $x = 4$.
Then you go back to your equation for $y$ to find the corresponding $y$-values for each solution $x$.
2x-y-7=0 Xy=4
Y=4/x
2x-4/x-7=0
Take lcm
2x²-4-7x=0
2x²-7x-4=0
(2x+1)(x-4)=0
X=-1/2, x=4
If x=-1/2
Y=-8
If x=4
Y=1
To answer the more theoretical question: Substitution and elimination are each valid techniques, and either will work for any problem. It is just that for larger problems, elimination tends to get less messy, and elimination sets you up for learning Gaussian Elimination (row-reduction) if you ever have to take matrices. Once you take Linear Algebra, you will understand why each of these methods has a place. :)
