How do we make sure the solution we find for an equation is correct? Accidentally adding when we meant to subtract, missing a negative when we distribute, forgetting to write an $x$ from one line to the next–there are many possible mistakes to watch out for!

Fortunately, each step we take solving an equation results in a new equation with the same solution as the original. This means we can check our work by substituting the value of the solution into the original equation. For example, say we solve the following equation:

\(\begin{align} 2x&=\text-3(x+5)\\ 2x&=\text-3x+15\\ 5x&=15\\ x&=3 \end{align}\)

Substituting 3 in place of $x$ into the original equation,

\(\begin{align} 2(3) &= \text-3(3+5)\\ 6&= \text-3(8)\\ 6&=\text-24 \end{align}\)

we get a statement that isn't true! This tells us we must have made a mistake somewhere. Checking our original steps carefully, we made a mistake when distributing -3. Fixing it, we now have

\(\begin{align} 2x&=\text-3(x+5)\\ 2x&=\text-3x-15\\ 5x&=\text-15\\ x&=\text-3 \end{align}\)

Substituting -3 in place of $x$ into the original equation to make sure we didn't make another mistake:

\(\begin{align} 2(\text-3) &= \text-3(\text-3+5)\\ \text-6&= \text-3(2)\\ \text-6&=\text-6 \end{align}\)

This equation is true, so $x=\text-3$ is the solution.