Problem 74 Simplify each expression. $$ ... [FREE SOLUTION] (2024)

Short Answer

Distributive Property

The distributive property is essential when you want to simplify expressions, particularly those involving parentheses. It means you multiply a single term by each term inside a set of parentheses. For example, if we have an expression like \(a(b + c)\), we distribute 'a' to both 'b' and 'c', making it \(ab + ac\). In the given exercise, we have an expression: \(5 - 6t - (3t + 4)\). Here, there's a negative sign outside the parentheses. We treat this negative sign like multiplying by -1. When we distribute, it changes the signs of the terms inside the parentheses. Using the distributive property, \(5 - 6t - (3t + 4)\) turns into \(5 - 6t - 3t - 4\). This step clears out the parentheses, making it simpler to move on to the next steps.

Combining Like Terms

Combining like terms is another crucial concept in simplifying algebraic expressions. Like terms are the terms that have the same variable raised to the same power. For instance, \2x\ and \5x\ are like terms because both consist of the variable 'x'. When combining them, you simply add or subtract their coefficients. In the exercise, after distributing the negative sign, our expression becomes: \(5 - 6t - 3t - 4\). We identify two groups of like terms here: \ -6t\ and \ -3t\, then \ 5\ and \ -4\. First, we combine the t terms: \ -6t - 3t\, which gives us \ -9t\. Next, we combine the constant terms: \ 5 - 4\, which results in \ 1\. Therefore, combining like terms, the expression simplifies to \ 1 - 9t\.

Negative Sign Distribution

The distribution of a negative sign can sometimes be confusing but is straightforward once you get the hang of it. When you have a negative sign outside of parentheses, you need to distribute it to each term inside. Think of it as multiplying by -1. This changes the sign of every term within the parentheses. In the example given: \(5 - 6t - (3t + 4)\), the negative sign in front of \(3t + 4\) implies \-1(3t + 4)\. Distributing the negative sign, we get \ -3t\ and \-4\, transforming our expression to \5 - 6t - 3t - 4\. This step is vital as it allows you to remove the parentheses and makes it easier to combine like terms.