91Ó°ÊÓ

Algebraic manipulation is a fundamental technique used in solving equations and simplifying expressions. In algebra, you basically play a balancing game – what you do to one side of the equation, you must do to the other to maintain equality. Starting with the equation \(A=\frac{5}{2}(B+3)\), you need to manipulate it algebraically to solve for \(B\). This involves distributing the fraction, subtracting terms, and dividing, all while making sure the equation remains balanced.

The process begins with distributing the fraction across the parenthesis, which means multiplying each term inside the parenthesis by \(\frac{5}{2}\). This step simplifies the equation and gets rid of the parenthesis, a crucial move before you can isolate the variable. Through careful manipulation - addition, subtraction, multiplication, and division - you can unravel the value of \(B\) and make it the subject of the formula.

Isolating the variable, \(B\) in this case, means getting \(B\) alone on one side of the equation. To achieve this, subtraction is first employed to eliminate the addition of \(\frac{15}{2}\) to \(B\). It's like peeling layers off to reveal what's underneath; you remove anything that is not part of the core variable term.

After subtracting \(\frac{15}{2}\) from both sides, the equation \(A - \frac{15}{2} = \frac{5}{2}B\) is obtained. The final act of isolation is dividing both sides by \(\frac{5}{2}\), which leaves \(B\) by itself. This step reflects a core principle of algebra: the goal is to single out the variable we're interested in, setting it free from other numerical and variable companions.

The distributive property is a tool that lets you multiply a single term by each of the terms inside a parenthesis. This is exactly what's happening in step 1 of the exercise, where \(\frac{5}{2}\) is distributed over \(B+3\). Mathematically, the expression \(\frac{5}{2}(B+3)\) becomes \(\frac{5}{2}B + \frac{5}{2} \times 3\) or \(\frac{5}{2}B + \frac{15}{2}\), after applying the distributive property.

As you might notice, distributing the fraction across the parenthesis transforms the equation into a more straightforward format without changing its meaning or value. The distributive property is a powerful ally when dealing with equations that involve terms grouped within parentheses and is essential for simplifying complex expressions.