7.5 Solving Equations

Having written down some complicated equation such as \[2x^2 + \frac{16}{x} = 0\] like we practiced in Question 7.1.9, we might want to know what number $x$ is. This is called ‘solving’ the equation. (In this case the solution is $x = -2$.) Or we might want a formula for some variable in terms of another. In mathematics it is often very useful to start off working on a much simpler problem, so let’s think of an equation that might be easy to solve so we can learn how to solve more difficult equations.

Suppose we know that $3x + 2 = 11$. This is the same as saying I’ve taken a number (which I’ve decided to call $x$), I’ve multiplied it by $3$, added $2$, and I got $11$. You want to work out what the number is. We’ve already done problems like this (see Question 4.1.7): if we want to know what we got after multiplying by $3$, but before adding $2$, we need to subtract $2$ from $11$, to undo the addition. We could write this down as: \[3x = 11-2.\] So now we know that $3$ times the number is $9$. Then we need to divide by $3$ to undo the multiplication, and we find that the number is $9\div 3 = 3$. The whole process could be written down as: \begin{align*} 3x + 2 &= 11\\ 3x &= 11-2\\ 3x &= 9\\ x &= 9\div 3\\ x &= 3. \end{align*}

You’ll notice that we end up writing the same thing on the lhs of the equation as we simplify the rhs: we’ve written ‘$3x = \cdots$’ and ‘$x = \cdots$’ twice. This can get a bit annoying, so mathematicians have decided to just leave the lhs blank if it’s the same as in the line above. So we write it like this: \begin{align*} 3x + 2 &= 11\\ 3x &= 11-2\\ &= 9\\ x &= 9\div 3\\ &= 3. \end{align*}

Now, it’s possible we’ve made a mistake somewhere, so it’s a good idea to go back and check: if $x=3$, does $3x + 2$ really equal $11$? Or in technical terms, does $x=3$ ‘satisfy’ the equation $3x+2=11$? (Yes, $3\times3 + 2 = 11$, so we haven’t made a mistake.)

Some equations have no solution, meaning that the equation can’t be true, for example $a = a+1$ has no solution because no matter what number you pick for $a$, the rhs will be $1$ more than the lhs, so they can’t be equal. Some equations have more than one solution, for example $x^2 = 1$ has two solutions: $x=-1$ or $x=1$.

Let’s take a closer look at how we solved that equation. How did we get from $3x = 9$ to $x = 9\div 3$? It’s almost as if the $3$ has moved from one side of the equation to the other. In fact, we wanted to get $x$ alone on the lhs, so it makes sense that we’d want to get rid of the $3$. And to get rid of it, we have to divide by $3$, knowing that $3x\div 3 = x$. If we want to divide the lhs by $3$, then we also have to divide the rhs by $3$.

Why should it make sense to divide both sides of the equation $3x = 9$ by $3$? An equation is just a statement that the expression on the left ($3x$) comes out to be the same number as the expression on the right ($9$); they’re two different ways of expressing the same number. If we take that number and divide it by $3$ then we get a new number, which we can express in two different ways: in terms of the lhs ($3x\div3$) or the rhs ($9\div 3$). So we get a second equation which is true as long as the first one is true.

Of course, this doesn’t only work for dividing by $3$; it works for anything you can do to a number: multiplying by another number, adding another number, raising to a power, taking a logarithm, etc. I call these things ‘actions’. And when I say ‘multiplying by a number’, the number can be any expression that represents a number – we can multiply both sides of an equation by $x^2 + 2$ if we want to. Usually we want to use an action that’s going to cancel with something on one side of an equation, to move it over the other side, in effect.

If you apply an action to both sides of an equation, you get a second equation which is true if the first equation is true. This allows you to move something from one side of an equation to the other by doing the opposite action.

Let’s take another look at solving $3x + 2 = 11$ using this idea. We want to end up with an equation that says $x=\text{something}$, so we’d like to move the $2$ and the $3$ over to the rhs. First we subtract $2$ from both sides: \[3x + 2 - 2 = 11 - 2.\] Then, simplify: \[3x = 9.\] Now we want to move the $3$ over the other side, so we divide both sides by $3$: \[3x\div 3 = 9\div 3\] and simplify: \[x = 3.\]

Apart from solving equations, there is another use for the techniques you will learn in this section. Sometimes you might be given a rule you can use on an equation of a certain form, and you might have to transform an equation into this form. For example, you might read something like ‘the equation $x^2 + y^2 = r^2$ represents a circle of radius $r$’, and you have an equation $x^2 - 4 = -y^2$ and you want to know whether it ‘represents a circle’ and if so, what is the ‘radius’. If you take that equation and add $y^2 + 4$ to each side, then simplify, you get $x^2 + y^2 = 4$, which now looks very much like the equation in the rule, but with $4$ instead of $r^2$. Since $4$ is $2^2$, you can conclude that the equation represents a circle of radius $2$.