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Cross-multiplication is a powerful technique for solving proportions. A proportion is an equation that states two ratios are equivalent. For example, in the equation \ \( \frac{a}{-8} = \frac{-42}{48} \) both fractions are equal. To solve this, we use cross-multiplication. To cross-multiply, you multiply the numerator of one fraction by the denominator of the other fraction. In our example, we multiply \ \(a \) by \ \(48 \) and \ \(-8 \) by \ \(-42 \). This results in the equation: \ \(48a = -8 \times -42 \). This way, we have simplified the problem from dealing with fractions to a more straightforward algebraic equation.

Simplifying fractions is an essential step in solving algebraic equations efficiently. After you cross-multiply your proportions, you would likely end up with a product that can be simplified. In our exercise, we started with \ \( 48a = 336 \). To isolate \ \(a \), we need to divide both sides by 48. The division process is straightforward because 336 divides evenly by 48. If the division involves more complex numbers, breaking them down into their prime factors can help simplify them. For instance, \ \(a = \frac{336}{48} \) can simplify to \ \(a = 7 \), making it more manageable and producing an exact result.

Basic algebra involves manipulating equations to solve for unknown variables. In our exercise, we initially set up the equation \ \( \frac{a}{-8} = \frac{-42}{48} \), and used cross-multiplication to form a simplified equation \ \(48a = 336 \). At this stage, basic algebra principles—such as isolating the variable by performing the same mathematical operations on both sides of the equation—come into play. After you simplify the multiplication, the next step is to isolate your variable (\(a\)). We achieve this by dividing both sides of the equation by 48. The final step involves simplifying these fractions to find \ \(a = 7 \). Mastering these basic algebra techniques allows you to tackle more complex mathematical problems with confidence.