Have you seen a numeric trick? Here are the steps of a numeric trick:

(i) Choose any number.

(ii) Multiply it by 6.

(iii) Subtract 21 from the result.

(iv) Divide this new result by 3.

(v) Subtract twice the number you chose from this last result.

(a) Try this sequence of five steps three times, starting each time with a different number. What was the result of your experiment?

(b) Next, using the letter $x$ to represent the number you chose in the first step, show why the results from part (a) are not just a coincidence, but a mathematical fact.
(a) Let's do the experiment with the numbers 0, 5, and -4.

$$
\begin{aligned}
& 0 \xrightarrow[\times 6]{ } 0 \xrightarrow[-21]{ }-21 \xrightarrow[\div 3]{ }-7 \xrightarrow[-(0 \times 2)=0]{ }-7 \\
& 5 \xrightarrow[\times 6]{ } 30 \xrightarrow[-21]{ } 9 \xrightarrow[\div 3]{ } 3 \underset{-(5 \times 2)=-10}{\longrightarrow}-7 \\
& -4 \xrightarrow[\times 6]{\longrightarrow}-24 \xrightarrow[-21]{ }-45 \xrightarrow[\div 3]{-15 \xrightarrow[-(-4 \times 2)=+8]{\longrightarrow}}-7
\end{aligned}
$$

The final result is always -7.

(b) It is reasonable, then, to conjecture that for any initial number chosen, the final result of this procedure will always be -7. Let $x$ be the initial number. We then have the following operations.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-040.jpg?height=88&width=1345&top_left_y=1658&top_left_x=390)

Therefore, the result of this "magic" will always be equal to -7, regardless of the number initially chosen.