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Cross-multiplication is a useful technique to eliminate fractions from an equation, making it easier to solve. This method involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa, essentially swapping and multiplying. This allows us to compare two products instead of dealing with fractions.
When applied to the equation \(\frac{a-6}{12}=\frac{a-2}{4}\), this method helps us make the fractions disappear so we can deal with a simpler equation. Specifically, you'll multiply \(a-6\) by 4 and \(a-2\) by 12. This gives the equation

• \((a-6) \times 4 = (a-2) \times 12\)

By doing so, we get rid of the fractions. Solving for the variable now becomes more intuitive as it resembles a basic linear equation. Cross-multiplication is thus a powerful first step to simplify fraction-based equations.

The distributive property is a fundamental algebraic principle that allows us to multiply a number by a group of numbers added together, distributing the multiplication across each term inside the parentheses. In our equation, we use this property to open up the parentheses and simplify the equation.
For example, after cross-multiplying, our equation is\(4(a-6)=12(a-2)\). Here, apply the distributive property:

• \(4 \cdot a - 4 \cdot 6 = 12 \cdot a - 12 \cdot 2\)
• This simplifies to \(4a - 24 = 12a - 24\)

This property aids in breaking down complex expressions into manageable pieces, key in solving equations. Notice how applying the distributive property helped transform the equation into a simpler form that is more straightforward to solve.

Once you have simplified the equation using cross-multiplication and the distributive property, the next step is solving for the variable, which, in our case, is \(a\). With the equation \(4a - 24 = 12a - 24\), we can simplify further by eliminating like terms on both sides.
First, notice that both sides contain \(-24\). Adding 24 to both sides cancels these terms out:

• \(4a - 24 + 24 = 12a - 24 + 24\)
• Which simplifies to \(4a = 12a\)

To isolate \(a\), subtract \(4a\) from both sides:

• \(4a - 4a = 12a - 4a\)
• Reducing to \(0 = 8a\)

Finally, divide both sides by 8 yielding \(a = 0\). However, it’s critical in problem-solving to check your solution by substituting back into the original equation. Here, both sides equate to \(\frac{-1}{2}\), showing the solution is valid for this linear equation, emphasizing thorough checking in algebraic solutions.