Repeat the following procedure for the four given numbers. Multiply the number by 8 . Add 12 to the product. Divide this sum by 2 . Subtract 6 from the quotient. The 1 st number is 2 . The result is 8 . The 2 nd number is 5 . The result is 20 . The 3rd number is 9 . The result is 36 . The 4 th number is 11 . The rult is 44 . a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as \( n \). The result is \( 4 n \). (Simplify your answer.) b. Represent the original number as \( n \), and use deductive reasoning to prove the conjecture in part (a). Multiply the number by 8 .

Multiplying the number \( n \) by 8 gives us \( 8n \). Next, we add 12 to this product: \( 8n + 12 \). Then, we divide the sum by 2: \[ \frac{8n + 12}{2} = 4n + 6. \] Finally, we subtract 6 from this quotient: \[ 4n + 6 - 6 = 4n. \] Thus, the result after performing all the operations on any number \( n \) is \( 4n \), which confirms our conjecture! Furthermore, this simple sequence of operations mirrors similar mathematical transformations frequently used in algebra. The linear relationship represented in the conjecture shows how systematic manipulations can lead to predictable results, a concept that becomes indispensable in various real-life scenarios, such as budgeting, scaling recipes, or even predicting outcomes based on initial inputs. For anyone interested in exploring this further or finding more mathematical patterns, delving into algebraic functions and transformations can be enlightening! These subjects not only reveal the underlying mechanics but also showcase how mathematics connects to the world, thus enhancing our analytical skills and problem-solving toolkit.