Canonical Form-Seeking: Isolating a Variable or Expression

Algebra is a way of solving for unknown values, called variables, in equations. Here are some things that you should know about solving algebraic equations:

$\bullet$ When you perform an operation (for example, $+,-,\times,\div$) on one side of an equation, you must do it to the other as well.

$\bullet$ There must be at least as many unique equations as there are variables for the equations to be solvable.

Here's an example:

Solve for $x$.

$x-2=6$

The way to go about solving something like this is to isolate the variable ($x$) on one side of the equation. We can do this by adding $2$ to each side. When we do this, we end up with:

$x=8$

We now know that $x$ is equal to 8.

Let's try another one:

Solve for $y$:

$7y - 5 = 9$

Add $5$ to both sides of the equation:

$7y = 14$

Now divide both sides of the equation by $7$:

$y = 2$

Solve for $x$.

$5x-1=2x+14$

This one is trickier.

To start, we add $1$ to both sides.

$5x=2x+15$

We can now subtract $2x$ from each side to get:

$3x=15$

To finish it off, we can divide both sides by $3$ to get that:

$x=5$

Isolation is not only a useful method for separating variables from an equation, but they can also be used to solve functional identities. Let's look at an example.

Often we get equations with functions of numerous variables. These can be very hard to deal with! Let's say we get an identity like:

$yf(x) = xf(y)$

What is the set of numbers that are solutions for f(x)? Substituting in numbers for x and y doesn't really get us a definite answer. So let's try isolating (there it is again!) the variables to their own separate sides.

We get:

(I)$\frac{f(x)}{x} = \frac{f(y)}{y}$

Now both sides of the function look pretty similar. This looks like something we can make into a new function!

We can define the function (II) $g(t) = f(t)/t$. So now, if we look at equation I, we can say that:

$g(x) = g(y)$

for any x and y.

Now think about all pairs (x, y). This function can only work if g(x) is a constant! So now let's go back to equation (I):

$\frac{f(x)}{x} = c$

so $f(x) = cx$ for all constants c is the family of solutions.

As you can see, isolation works pretty well for functions too, not just variables!