91Ó°ÊÓ

In algebra, the distributive property helps us to simplify expressions by multiplying each term inside the parentheses by the factor outside. For example, in the expression \(0.42(2x)\), you distribute by multiplying 0.42 by 2x to get 0.84x. Similarly, for \(0.54(80 - 3x)\), you multiply 0.54 by both 80 and -3x, resulting in \(43.2 - 1.62x\). This step is crucial for breaking down complex expressions, so you can eventually combine like terms easily.

Combining like terms means adding or subtracting terms that have the same variable raised to the same power. In our example, after distributing, we have the expression \(0.35x + 0.84x + 43.2 - 1.62x\). To combine like terms, focus on those with the variable 'x': \(0.35x, 0.84x,\) and \(-1.62x\). Add these coefficients together: \(0.35 + 0.84 - 1.62\), which simplifies to \(-0.43x\). Finally, combine to form a single term: \(-0.43x\). This step simplifies the expression significantly, making it easier to work with.

Simplifying algebraic expressions involves systematically applying a series of steps.
1. **Distribute:** This is the first step, where you use the distributive property to remove parentheses, allowing you to break down the expression.
2. **Rewrite the Expression:** After distributing, rewrite the expression to clearly see all terms. For instance, after distributing in \(0.42(2x)\) and \(0.54(80 - 3x)\), you get \(0.35x + 0.84x + 43.2 - 1.62x\).
3. **Combine Like Terms:** Gather terms with the same variables and combine them. In our case, combine \(0.35x, 0.84x,\) and \(-1.62x\) to get \(-0.43x\).
4. **Simplify:** Finally, include any constant terms that remain. After combining like terms in our example, you get the final expression \(43.2 - 0.43x\). Following these steps ensures clear and accurate simplification.