There is a right way to think about equations and there is a wrong way to think about equations. Let us not speak of the wrong way…

The equations I have in mind are those equations written in the form

$f(x)=g(x)$,

where the aim is to find all the real numbers $x$ that ‘satisfy’ the equation.

We aren’t always taught the logic behind solving equations. The first thing to say is that many of us are trained to believe that this ‘$=$‘ means the ‘the answer is’. This is not what equals means. This may have happened to us because while young children our textbooks had stuff like

$2+6=\dots$

written in them… the ‘answer’ of course being eight and the = sign almost suggests that we have to ‘do something’ to $2+6$. Of course, this is not what equals means, and while the pupil who writes

$2+6=8$

is correct, the pupil who writes e.g.

$2+6=11-3$,

has written a statement just as true as $2+6=8$.

Equals, in this context, signifies that two numbers are the same… or three numbers are the same:

$56\%=0.56=\frac{14}{25}$.

It is also a consequence of the way we read English that we say things like ‘we have $x+2$ on the left and 4 on the right’ when we see

$x+2=4$.

In reality there is no ‘left’ and ‘right’ just a pair of numbers that are the same number. This means that to solve an equation of the form

$f(x)=g(x)$,

what we really want to do is find real numbers $x_1,x_2,\dots$ such that $f(x_i)$ and $g(x_i)$ are the same number.

We do usually want to end up with, where there is a unique solution $x_1$,

$x=x_1$,

but this is a function of how we read more than anything. It is just as valid to write

$x_1=x$.

The one thing we are never told when we meet equations is the logic of how this works… the first thing we do, although we never explicitly say it, is say:

Assume $x$ satisfies the equation.

What we do from here on is make a series of implications about the number $x$ until we are forced to conclude that $x$ must be equal to, say, two. Let us demonstrate this with an example.

Example

Solve $3x-2=13$.

Solution: Assume that $x$ has the property that

$3x-2=13$

If these numbers are equal, then adding two to both of them will yield two more numbers that are still equal:

$3x-2+2=13+2$.

Of course, $-2+2=0$, $3x+0=3x$ and $13+2=15$ so that $x$ has the property that

$3x=15$.

If these numbers are equal, then multiplying both of them by one third will yield two more numbers that are still equal:

$\frac{1}{3}(3x)=\frac{1}{3}(15)$.

Of course, $\frac{1}{3}(3x)=x$ and $\frac{1}{3}(15)=5$ and so we are led to conclude that $x=5$ $\bullet$

Here we see that by assuming that $3x-2=13$, we were forced to conclude that $x=5$.

Suppose that $S_1$ is a statement. For example, $S_1$:=”Today is Wednesday”. If a statement $S_2$ is true whenever $S_1$, then we write $S_1\Rightarrow S_2$. For example,

“Today is Wednesday” $\Rightarrow$ “Today is a Weekday”.

This symbol $\Rightarrow$ has this meaning and is called “implies that”. Of course, if

$A\Rightarrow B\Rightarrow C$ then $A\Rightarrow C$,

and so we can build up a chain of ‘implications’ and then cut it down to size. This is what we did above in our example:

$x$ satisfies $3x-2=13$

$\Rightarrow 3x=15$

$\Rightarrow x=5$,

so “$x$ satisfies the equation” $\Rightarrow x=5$.

This is the basic logic behind solving equations. In a later post we will explore how to use this logic to strategically solve equations.