A.2: Basic algebra
The field of algebra gives us the basic tools we need to manipulate equations, or to pull useful information out of a mathematical sentence. There is one all-encompassing rule that must be obeyed:
Whatever you do on one side of an equals sign, you must do on the other!
If you multiply the left side of an equation by eight, you must multiply the right side by eight. If you subtract twelve from the left side, you must subtract twelve from the right side. You have no choice. This is fundamental to the logic system of equations.
This rule of algebra applies just as well to letters as it does to numbers—you can replace any number in an equation with a letter and still perform algebraic steps.
Example a.1
Solve the equation
\[ \frac{z + x}{y} = w \]
for \(x\).
This is exactly the same as the example in the previous section, but with letters instead of numbers. We will follow the same steps to solve for \(x\):
\[
\begin{align*}
\frac{z+x}{y} &= w \\
\left(\frac{z+x}{y}\right)(y) &= w(y) \\
z + x &= w(y) \\
z + x – z &= w(y) – z \\
x &= w(y) – z
\end{align*}
\]
Even though there were no numbers involved, we can still find a solution for \(x\). Because it is all symbols,* we call this a symbolic solution.
You can check that this is the correct answer by returning to the original equation and substituting in our solution:
\[
\begin{align*}
\frac{z + x}{y} &= \frac{z + \left(w(y) – z\right)}{y} \\
&= \frac{z + w(y) – z}{y} \\
&= \frac{w(y)}{y} \\
&= w
\end{align*}
\]
Good! Our solution is correct!
*When you get down to it, a number is also just a symbol. Whether we write the number 8 or the letter \(y\), we’re just writing some squiggly line on a piece of paper to convey some meaning.