If \(\frac{x}{3}=4\) and \(x+y=32\), what is the value of \(x-y ?\) A) \(-24\) B) \(-8\) C) 12 D) 32

In the world of mathematics, an algebraic equation is like a puzzle waiting to be solved, where variables and constants interact through operations such as addition and subtraction. These equations represent a statement of equality, with the 'equals' sign (\(=\)) serving as a scale that needs to balance both sides.

For example, the equation given in our exercise, \frac{x}{3}=4\, creates a scenario where one-third of a certain number equals four. To find this mysterious number, we must perform the inverse of dividing by three—multiplying by three. This transforms our equation into a more straightforward expression: \(x = 4 \times 3\), which reveals that \(x\) is 12. Being able to maneuver these equations is vital for unraveling puzzles in algebra and beyond.

The substitution method is an elegant dance where one variable leads, and the others follow. It's a strategy employed in algebra to find the values of variables within a system of equations. In substitution, we use one equation to find the value of one variable and then 'substitute' this value into another equation.

Consider our scenario: once we've determined that \(x = 12\), we can substitute this value into the second given equation, \(x + y = 32\), replacing \(x\) with 12: \(12 + y = 32\). It's like telling a story where one character's actions influence the outcomes of another. By following the substitution method, the value of \(y\) is no longer a mystery, it's simply the result of 32 minus 12, which is 20.

The four arithmetic operations are the foundation of mathematics. They consist of addition (+), subtraction (-), multiplication (\(\times\)), and division (\(\div\)). When we apply them correctly, they help us solve various mathematical problems with precision and ease.

In our SAT math problem, we see these operations in action. First, multiplication aided us in moving from \(\frac{x}{3} = 4\) to \(x = 4 \times 3\). Next, subtraction was the key to finding the value of \(y\), with \(y = 32 - 12\). Finally, to calculate \(x-y\), we combine the values of \(x\) and \(y\), which were a result of the earlier steps, through subtraction: \(12 - 20\), leading us to the solution -8. Through these operations, arithmetic becomes the silent hero of problem-solving.