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When solving linear equations, the distributive property is a crucial tool. It allows you to remove parentheses by distributing (multiplying) a constant across the terms inside the parentheses. For example, in the equation given, the left side is \(3(6 - 4x)\). By applying the distributive property, you multiply 3 by each term inside:

• 3 times 6 equals 18,
• 3 times -4x equals -12x.

This gives us the expression: \[3(6 - 4x) = 18 - 12x\] Similarly, on the right side, applying the distributive property gives us: \[2(-6x + 9) = -12x + 18\] This property simplifies complex equations, making them easier to solve.

Combining like terms is another fundamental step in solving equations. After using the distributive property, it’s time to simplify the equation further. 'Like terms' are terms that have the same variable raised to the same power. In our example, once distributed, we have: \[18 - 12x = -12x + 18\]
Notice both sides have \(-12x\) and constants (18). Because the left and right sides are identical, any value of \(x\) will keep the equation balanced. However, if the terms weren't identical, one would combine like terms on the same side of the equation to consolidate the equation further. Remember, combining like terms helps reduce the equation to simpler forms, thus making it easier to handle.

Verifying your solution is always a good practice. It ensures that you have correctly solved the equation and that your solution satisfies the original equation. For our problem, because both the sides turned out identical: \[18 - 12x = -12x + 18\]
This means the equation is true for all real numbers. We can select a simple value, like \(x = 0\), to check:

• On the left side: \(3(6 - 4 \times 0) = 3 \times 6 = 18\)
• On the right side: \(2(-6 \times 0 + 9) = 2 \times 9 = 18\)

Since both sides equal 18, our original solution holds. Always substitute your solution back into the original equation to confirm its correctness. This verification step cements your confidence in your problem-solving skills.