Expressing a variable
How to express one particular variable from a complex relationship or fraction. This technique is especially used when working with different formulas in which multiple variables are involved. Yet in combination with other formulas we can achieve unprecedented feats where we derive everything from nothing.
Simple linear expressions
Consider this pattern: ax + b = c. How do we isolate x from this imaginary pattern ? We always start by putting all the expressions with the variable we are isolating on the left-hand side and the rest on the right-hand side. In theory it doesn't matter which side what is on, but the convention is to have the unknown we are expressing on the left. So first we move b to the right side, that is, we add −b to the equation:
\begin{eqnarray} ax + b &=& c \quad /-b\ ax &=& c -b \end{eqnarray}
We have expressions with the variable on one side, but we still have that a bothering us. How to get rid of it elegantly? What do we need to do with the expression ax to get only x? Yes, we divide the expression - and therefore the whole equation - by a, or multiply by $\frac{1}{a}$. The value of a must be different from zero(we can't divide by zero). If we do so, we get:
\begin{eqnarray} ax &=& c -b\quad\cdot\frac{1}{a}\quad(a\ne0)\x &=& \frac{c-b}{a} \end{eqnarray}
Done. Another example:
$$2ax − 3bx = 10a$$
Again we have to isolate x. We already have the expressions with the unknown on the left-hand side, so we have a bit of work saved. But the complication is that we have two expressions with unknown. We solve the situation by reproaching. From the left-hand side of the equation, we print x and get:
\begin{eqnarray} 2ax - 3bx &=& 10a\\x(2a - 3b) &=& 10a \end{eqnarray}
When we had the expression ax in the previous example and wanted to get just x, we multiplied the whole equation by $\frac{1}{a}$. Now we have the expression x(2a − 3b) in the equation and again we want to know x. We do the same thing, we divide the whole equation by the expression (2a − 3b). Don't worry, we can indeed divide the equation by the whole bracket, it just has to be 2a − 3b ≠ 0.
\begin{eqnarray} x(2a - 3b) &=& 10a\quad /\cdot \frac{1}{2a - 3b}\x &=& \frac{10a}{2a-3b} \end{eqnarray}
Fractions
If we have an expression with fractions, we use the same equation modifications, we just have to be careful not to accidentally divide or multiply by zero. Example:
$$ \frac{a}{x}=\frac{b}{c} $$
We want to isolate x. We multiply the whole equation by x - we just have to add the assumption that x ≠ 0. This is true anyway, because the fraction $\frac{a}{x}$ is in the equation and x is in the denominator - we can't divide by zero. So we get the equation:
\begin{eqnarray} \frac{a}{x}&=& \frac{b}{c}\quad/\cdot x\quad(x\ne0)\& &=& \frac{bx}{c}\quad/\cdot c\quad(c\ne0)\& ac&=& bx\quad\cdot\frac{1}{b}\quad(b\ne0)\\frac{ac}{b}&=& x&=&\frac{ac}{b}\end{eqnarray}
When isolating the unknown, simply use the usual equational equation equations.