Basics of Algebra: Part II

by Oren Lahav

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:

Equations are equivalence relations between 2 numbers, variables or expressions, for example x=2y

. As we've already seen, equations such as 3x=6

can be simplified to their simplest form, x=2

, in which we can identify the number our variables represent. In this case, our number x represents 2.

Note that this doesn't work with simple expressions. If we just have the expression 15x+22

, there's nothing we can say about x. It's the "=" symbol that allows us to say something about x.

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. 2x+1=2x+2 is a good example, since when simplifying we get 1=2 , which is clearly never true. On the other hand, some equations are true for all values of x. For example, x+2=x+2 , which simplifies to 0=0 , 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 $4-3x \ge 7$ . Then while simplifying we subtract 4 from each side, $- 3x \ge 3$ and finally divide by 3, flipping the sign: $x\le- 1$ . Plugging this into the original inequality shows how important flipping the sign is- plug in -2, which satisfies the solution, to get $10\ge7$ , which is true. If you forget to flip the sign you get $x\ge-1$ , and plugging in 2 for example will result in $-2\ge7$ , 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 x^2 . That we'll do next time, but for now, let's throw in another variables.

Equations in two variables look a lot like equations with 1 variable. Say x+y=1

. Can you solve for x and y? No, you can't. There are too many possibilities- x=1

and

works, but so do x=2

and

. There are actually infinitely many possible combinations. In reality, this equation represents a line in the XY plane, but that's an analytic geometry thing, we're doing algebra.

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 x+y=2

and

. A simple guess and check shows x=1

and

. But there are more formal ways of solving equations in two unknowns.

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 x=y , and substitute this into equation 1: y+y=2 , 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 2x=2 , and we can solve from there.

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.