Find x if 54 divided by (x - 7) equals 5 squared minus 4 squared.

Step-by-step solution

1. 1

   Evaluate the right-hand side. $5^2-4^2 = 25-16 = 9$. (Equivalently, by the difference of squares, $(5-4)(5+4) = 1\cdot9 = 9$.) The equation is now $54 : (x-7) = 9$.

2. 2

   Notice that the unknown is the divisor, not the dividend. In $a : b = c$ the missing divisor is recovered by $b = a : c$, not by multiplication. Getting this backwards would give $486$ instead of $6$.

3. 3

   Solve for the bracket. $x-7 = 54 : 9 = 6.$

4. 4

   Solve for x. $x = 6+7 = 13.$

5. 5

   Check the answer and the domain. $54 : (13-7) = 54:6 = 9$, matching $5^2-4^2=9$. Also $x-7 = 6 \neq 0$, so the division was legitimate - a divisor of zero would have made the original expression undefined.

6. 6

   Contrast with the dividend case. If the problem had read $(x-7):54 = 9$, the inverse step would be multiplication: $x-7 = 486$ and $x = 493$. Reading which side the unknown sits on decides the whole solution.