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Understanding the distributive property is essential for simplifying algebraic expressions. It's a tool that helps us multiply a single term by each term within a parenthesis. Imagine you have a basket of apples (the single term) and inside the basket, there are red and green apples (the terms inside the parenthesis). The distributive property would be like multiplying the basket's quantity of apples with both the red and green apples individually.

For example, in the expression \(\frac{1}{3}(x+6)\), applying the distributive property, we distribute \(\frac{1}{3}\) across \(x\) and \(6\):

• \(\frac{1}{3}\times x = \frac{1}{3}x\)
• \(\frac{1}{3}\times 6 = 2\)

This process is repeated for the second part \(\frac{4}{3}(x-9)\), where \(\frac{4}{3}\) is distributed similarly:

• \(\frac{4}{3}\times x = \frac{4}{3}x\)
• \(\frac{4}{3}\times -9 = -12\)

This step is fundamental and must be performed accurately to ensure the rest of the simplification process goes smoothly.

Once you've distributed your terms, the next step is combining like terms. This means adding or subtracting terms that have the same variable raised to the same power. Imagine having piles of coins; some are dimes, and some are nickels. When you're asked to find the total value, you would combine all the dimes together and all the nickels together, not mix them up.

In our algebraic expression, we find terms that involve \(x\) and those that are constants (just numbers). Then, we group them:

• Terms with \(x\): \(\frac{1}{3}x - \frac{4}{3}x\)
• Constants: \(2 + 12\)

By combining the terms with \(x\), we subtract \(\frac{4}{3}\) from \(\frac{1}{3}\), which gives us \(\frac{-1}{3}x\). And by combining the constants, we add \(2\) and \(12\) to get \(14\). This simplification of like terms results in a clearer and more concise expression \(\frac{-1}{3}x + 14\).

Algebraic expressions are phrases that contain numbers, variables, and operations. They're like puzzles, where each piece has to fit perfectly to show the complete picture. When we're given an expression like \(\frac{1}{3}(x+6)-\frac{4}{3}(x-9)\), the goal is to simplify it to reveal a simpler form that reveals the relationship between variables.

An expression doesn't equate to anything (there's no equal sign), it just exists on its own, representing a value that will vary depending on the value of \(x\). Simplifying it involves the curly dance of distributing and combining terms to choreograph a more elegant and easy-to-understand expression. In the end, the expression \(\frac{-1}{3}x + 14\) is as simple as it gets, telling us for any value of \(x\), what the expression will evaluate to without doing the full dance each time.