Algebra, part II- solving equations, inequalities, 2-variable equations
This series of lessons is designed to help you learn, or review, the fundamentals of algebra. In this lesson we continue our discussion of relations and extend this to solving equations and inequalities.
One of the most useful aspects of algebra is every day life is using algebraic expressions to model real life situations. This involves equations and their solutions.
We begin by reviewing
equations:
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Equations
are equivalence relations between 2 numbers,
variables or expressions, for example Note that this doesn't work with simple expressions. If we just have the expression |
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Of course, we can't say something about x all
the time. Some equations have no solution,
meaning no real number value for x satisfies the relation.
is a good example, since when simplifying we
get
,
which is clearly never true. On the other hand, some equations
are true for all values of x. For example,
, which simplifies to
, is always true
regardless of x.
Solving equations is easy, but what about inequalities?
Inequalities, <,>,etc., work the same way as equalities in terms of simplification, with one important difference. When we multiply or divide both sides by a negative value in inequivalence relations, we have to flip our inequality sign.
For example, say .
Then while simplifying we subtract 4 from each side,
and finally divide by 3, flipping the sign:
. Plugging this into the original inequality shows how
important flipping the sign is- plug in -2, which satisfies the
solution, to get
, which is true. If
you forget to flip the sign you get
, and plugging in 2
for example will result in
, which is
obviously wrong.
This is fun! Can we kick it up a notch?
Sure, if you insist, let's get to a more
challenging level. Until now we've dealt with linear equations
with one variable- i.e. we had only one variable and that
variables only had degree 1, so we had no . That we'll do next time,
but for now, let's throw in another variables.
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Equations in two
variables look a lot like
equations with 1 variable. Say In order to solve equations in 2 variables, we need at least 2 equations, as Elmo here clearly understands. This is analogous to finding the intersection of 2 lines in a plane. The same principle works for higher numbers- you have 5 unknowns? You need 5 equations. So, for our purposes, let's say |
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Substitution:
Writing one variable in terms of the other and
substituting it into the other equation. In our example, we can
rewrite equation 2 as , and substitute this into
equation 1:
, and solve from there.
Elimination:
Multiplying the equations by constants so that
we can add/subtract them from each other and eliminate one of the
variables. For example, if we add equations 1 and 2 together we
get , and we can solve from there.
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Is it really that
easy? Yes and no. No, because I can throw equations at you with really ugly irrationals, and it'll take you years to solve. But also Yes, because that really is all there is to it, the technique is simple. All you really need now is practice. Working with more than one variable introduces some other concepts, like factoring and expanding, which we'll talk about next time. We'll also finally get to quadratic equations and maybe some exponents. |
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Thanks for reading this Welcome to Algebra Lesson!
Click Here for Algebra-part-iii
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