Sometimes it's possible to look at the structure of an equation and tell if it has infinitely many solutions or no solutions. For example, look at

\(2(12x+18)+6=18x+6(x+7).\)

Using the distributive property on the left and right sides, we get

\(24x+36+6=18x+6x+42.\)

From here, collecting like terms gives us

\(24x+42=24x+42.\)

Since the left and right sides of the equation are the same, we know that this equation is true for any value of $x$ without doing any more moves!

Similarly, we can sometimes use structure to tell if an equation has no solutions. For example, look at

\(6(6x+5)=12(3x+2)+12.\)

If we think about each move as we go, we can stop when we realize there is no solution:

\(\begin{align} \frac16 \boldcdot 6(6x+5)&=\frac16 \boldcdot (12(3x+2)+12) &&\text{Multiply each side by $\frac16$.}\\ 6x+5 &= 2(3x+2) + 2 &&\text{Distribute $\frac16$ on the right side.}\\ 6x+5 &= 6x+4+2 &&\text{Distribute 2 on the right side.} \end{align}\)

The last move makes it clear that the **constant terms** on each side, 5 and $4+2$, are not the same. Since adding 5 to an amount is always less than adding $4+2$ to that same amount, we know there are no solutions.

Doing moves to keep an equation balanced is a powerful part of solving equations, but thinking about what the structure of an equation tells us about the solutions is just as important.