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Understanding the distributive property is crucial when solving linear equations. It allows you to multiply a single term by all terms within a parentheses. For example, if you have an equation like 3(2x - 1), you would apply the distributive property by multiplying 3 by each term inside the parenthesis: \( 3 \times 2x = 6x \) and \( 3 \times -1 = -3 \), resulting in \( 6x - 3 \). It is an essential step in simplification that prepares the equation for further solving.

In our exercise, \(3(2 x-1)-(6 x-4)=-9\), we used this property to distribute the 3 and the -1 (implied by the subtraction sign) across their respective terms. The distributive property transformed our problem into a simpler form, eliminating the parentheses and making the equation easier to work with.

Once you've distributed, the next step is to combine like terms. Like terms are terms in an equation that have the same variable raised to the same power. In simpler terms: look for terms that can be added or subtracted because they 'look alike'.
For instance, \(6x - 3 - 6x + 4 \) from the distributed equation means bringing together the \(6x\) terms and the constant numbers. In this case, \(6x - 6x = 0x\) which, in essence, nullifies the x term as anything multiplied by 0 vanishes. For the constants, \( -3 + 4 = 1\). This simplification, resulting in \(0x + 1\), sets up the stage for the last step in solving the equation.

If not performed correctly, combining like terms can lead to wrong answers, so this practice is about careful identification and manipulation of the elements in the equation.

Sometimes, after simplifying an equation using the distributive property and combining like terms, you'll find that it leads to a false statement. This situation indicates a 'no solution' equation, meaning there is no possible value of the unknown that can make the equation true.
In the given problem, we were left with the simple statement \(1 = -9\) after simplification. This is clearly a contradiction since 1 can never equal -9. Therefore, no value of x will satisfy the original equation - it's what we term as an equation with no solution. Realizing when an equation has no solution is important, as it tells us there are limitations to the problem or that a mistake might have been made earlier in solving the equation.

Understanding 'no solution' equations will significantly impact your problem-solving skills as it emphasizes the logic behind the mathematics, not just the manipulation of numbers and symbols.