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# Systems of Linear Equations

Systems of equations can be solved by a variety of methods. The simplest are:

Substitution

In the system of equations $$x+y = 5$$ and $$x-y = 3$$, we can solve for one of the variables and then substitute that relation into the other equation. From the second equation, we see that $$x = y+3$$, so we can rewrite the first equation as $$(y+3) + y = 5$$, thus giving us $$2y = 2 \implies y = 1$$. Thus $$y = 1$$ and $$x = 4$$.

Elimination

Using the same pair of equations as above, we could also combine the two equations together in such a way as to eliminate one of the variables. In this case, we can simply add them together. Thus we have $$(x+y) + (x-y) = 3 + 5$$, which gives us $$2x = 8 \implies x = 4$$. Thus $$y = 1$$ and $$x = 4$$.

Note by Arron Kau
3 years, 2 months ago