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The distributive property is essential for solving equations that involve parentheses. It states that multiplying a number by a sum (or difference) is the same as multiplying each addend individually and then summing (or subtracting) the products. For example, in the given equation: equation: \(-2(a-6)=4(a-3)\), we distribute \(-2\) and \(4\) to the terms inside the parentheses step by step. Multiplying \(-2(a-6)\) becomes: -2 \cdot a - 2 \cdot (-6) = -2a + 12\ and Multiplying \(4(a-3)\) becomes: \(4 \cdot a - 4 \cdot 3 = 4a - 12\). As a result, our equation after distribution becomes: \(-2a + 12 = 4a - 12\). Remember, this method ensures that each term inside the parentheses is correctly accounted for when simplifying an equation.

Combining like terms refers to the process of simplifying an equation by merging terms that have the same variable. After distributing in the equation \(-2a + 12 = 4a - 12\), the next step is to collect all terms containing the variable on one side and the constant terms on the other side. This forms a simpler equation that's easier to solve. Start by adding \(2a\) to both sides to transpose the variable term on one side: \(12 = 6a - 12\).
Then, add \(12\) to both sides to transpose the constant terms: \(24 = 6a\).

By doing this, we effectively simplify our original equation.

The final goal in solving a linear equation is to isolate the variable, meaning to get the variable alone on one side of the equation. From our earlier simplified term \(24 = 6a\), we isolate \(a\) by dividing both sides of the equation by 6: \[ a = \frac{24}{6} \]. Performing the division, we find that \(a = 4\). This step is critical because it gives us the solution for the variable. Always remember to perform the opposite operation to isolate the variable. If it’s multiplied by a number, you divide by that number, and if added, you subtract.