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When you encounter a proportion, like \(\frac{7}{21} = \frac{a}{18}\), cross multiplication is a powerful technique to solve for the unknown. It's a process that involves multiplying the numerator of one fraction by the denominator of the other. Then, you do the same for the other pair. Cross multiplication transforms a proportion into a simple equation. For our example, we would multiply 7 by 18 and 21 by a, resulting in \(7 \times 18\) and \(21 \times a\). This gives us a straightforward equation to solve: \(7 \times 18 = 21 \times a\). The technique is called cross multiplication because the multiplication occurs across the equals sign, forming an 'X' shape with the terms of the fractions. It's essential in ensuring both sides of the equation remain balanced.

Once we have the equation \(126 = 21a\), we need to solve it to find the value of \(a\). Solving equations is about finding the variable— in this case, \(a\). The goal is to isolate \(a\) on one side of the equation, which involves reversing the operations affecting the variable. Here, \(21a \) means \(a\) is multiplied by 21. To isolate \(a\), we need to perform the opposite operation. That means dividing both sides by 21: \(\frac{126}{21} = a\). This division simplifies to \(a = 6\). By solving the equation, we've determined that \(a\) is 6, ensuring the original proportion holds true.

Algebraic manipulation involves rewriting and simplifying equations or expressions to find the value of the unknown variable. In our case, we start with the proportion \(\frac{7}{21} = \frac{a}{18}\). Through cross multiplication, we rearrange this into the equation \(126 = 21a\). Next, we isolate \(a\) by performing algebraic operations: in this context, dividing both sides by 21. It's crucial to perform the same operation on both sides to maintain the balance of the equation. When we divide \(126 \) by \(21\), we simplify it to \(a = 6\). Mastering algebraic manipulation enables us to systematically approach and solve various mathematical problems, from simple to complex, by logically transforming equations step-by-step.