How does Solving Equations Work?¶

Let's talk about equivalence, also known as "if and only if", or $\Leftrightarrow$. It is a combination of $\Leftarrow$ and $\Rightarrow$, as you might expect:

• $A \Leftarrow B$ means that if $B$ is true, then $A$ is true.
• $A \Rightarrow B$ means that if $A$ is true, then $B$ is true.
• $A \Leftrightarrow B$ means that we have both $A \Rightarrow B$ and $A \Leftarrow B$; if one of $A$ and $B$ is true, then the other one is also true.

If one of $A$ and $B$ is false and we have $A \Leftrightarrow B$, then the other is also false; for example, if $A$ is false, then $B$ can't be true because that would imply that $A$ is also true.

At a higher level, $A$ and $B$ have the same "truth value", either both true or both false. This means that we can solve an equation like this: $$ \begin{align} \text{equation we want to solve} &\iff \text{something} \\ &\iff \text{something} \\ &\qquad \vdots \\ &\iff \text{something} \\ &\iff x = 5 \text{ or } x = 7 \end{align} $$ This guarantees that $x=5$ and $x=7$ are the solutions of the equation, and there are no other solutions; the original equation is true if $x$ is $5$ or $7$, and otherwise it's false. For this reason, we always solve equations so that we have equivalence at each step. It is very common to teach students to do this without writing down the equivalence signs.

Common things that can be done to an equality so that we get equivalence:

• Adding a number on both sides: $x=y \iff x+z=y+z$.
• Subtracting a number on both sides.
• Multiplying both sides by a nonzero number: if $z \ne 0$, then $x=y \iff xz=yz$. We need a nonzero $z$ for this, because we also want to say that if $xz=yz$, then $x=y$; this doesn't work if $z=0$. For example, $3 \cdot 0 = 4 \cdot 0$, but $3 \ne 4$.
• Dividing both sides by a nonzero number: if $z \ne 0$, then $x=y \iff \frac{x}{z} = \frac{y}{z}$.

I'll start by justifying the first item of the list, why adding a number on both sides gives equivalence. If $x = y$, then we of course have $x+z = y+z$, because $x=y$ means that $x$ and $y$ are the same; we can substitute $x$ for $y$ wherever we want. This proves the $\Rightarrow$ part of $\Leftrightarrow$, and for the $\Leftarrow$ part, suppose that $x + z = y + z$. Now we get $$ x = (x+z)-z = (y+z)-z = y, $$ where we substituted in the assumption $x+z = y+z$ in the middle step.

Other items of the list can be justified in a very similar way. For the $\Leftarrow$ part, we need something that cancels the effect of doing something on both sides. Above we canceled the addition of $z$ by subtracting $z$. We can similarly cancel a subtraction of $z$ with adding $z$, a multiplication by $z$ with dividing by $z$, or a division by $z$ with multiplying by $z$.

More generally, if $f$ is a function that has an inverse function, then we can apply $f$ on both sides of the equation, as in $$ x = y \iff f(x) = f(y). $$ Again, the $\Rightarrow$ part of $\Leftrightarrow$ is just substitution, and for the other direction, we can use the inverse function $f^{-1}$ as follows: if $f(x) = f(y)$, then $$ x = f^{-1}(f(x)) = f^{-1}(f(y)) = y, $$ where the first and last equality sign use the definition of inverse function, and the middle equality comes from substituting the assumption $f(x)=f(y)$.

The above rules are actually special cases of this with different functions; for example, the rule for adding $z$ on both sides uses the function $f(x)=x+z$, which has the inverse function $f^{-1}(x)=x-z$.