The difference of a number and 6 is the same as 5 times the sum of the number and 2. What is the number?
Let \(x\) represent the unknown number. According to the problem description, "the difference of a number and \(6\)" is written as \(x - 6\). Furthermore, "5 times the sum of the number and \(2\)" is written as \(5(x + 2)\). The problem states that these two expressions are equal, leading to the following linear equation:
\[x - 6 = 5(x + 2)\]
To solve for \(x\), we first expand the right side of the equation:
\[x - 6 = 5x + 10\]
Next, we isolate the variable \(x\). We can do this by subtracting \(x\) from both sides:
\[-6 = 4x + 10\]
Then, we subtract \(10\) from both sides:
\[-16 = 4x\]
Finally, dividing both sides by \(4\) gives:
\[x = -4\]
Upon reviewing the original equation and the provided options, it appears that the calculated value \(x = -4\) is not among the options listed (\(4, -2, -1, 1\)). Let us verify the calculation:
If \(x = -4\):
Left side: \(-4 - 6 = -10\)
Right side: \(5(-4 + 2) = 5(-2) = -10\)
The calculation \(x = -4\) is correct. Given the constraint of selecting the correct answer from the provided options, and noticing that no option matches the result of this interpretation, we conclude the answer is indeed \(-4\). As this is a multiple-choice question where the result is not present, we provide the correct calculated value. The result is \(\boxed{-4}\).
This type of algebraic translation is used in various real-life situations, such as budgeting or resource allocation. For example, if you are comparing two different payment plans where one plan is a flat discount and another is a percentage-based scaling, setting up an equation allows you to find the exact threshold where both plans cost the same. Mastering this skill helps in making informed financial or logistical decisions by clearly modeling constraints mathematically.