Problem 70 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-4-5(t-13)\)

Step-by-Step Solution

Answer

-5t + 61

Step 1: Distribute the Multiplication

Distribute the \(-5\) across the terms inside the parentheses. This means \(-5\) will be multiplied by \(t\) and by \( -13 \). Mathematically: \(-5(t-13) = -5t + 65\)

Step 2: Combine Like Terms

Now combine the constant term from the distribution with \(-4\). This means adding \(-4\) with \(65\). So \(-4 + 65\) simplifies to 61.

Step 3: Write the Final Expression

Now that \(-4\) and \(65\) are combined, write the final simplified expression: \(-5t + 61\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, helping us to break down and simplify complex expressions. In this property, a term is multiplied by each term inside a set of parentheses.

For example, let's take the expression \(-5(t-13)\). We distribute the \-5\ across the terms inside the parentheses. That means \(-5\) is multiplied by \t\ and \-13\.

Mathematically, this looks like:
\(-5(t) + (-5)(-13) = -5t + 65\).

Notice how multiplying \-5\ by \-13\ gives us a positive \65\ because multiplying two negative numbers together results in a positive number.

Combining Like Terms

Combining like terms is another crucial step in algebraic simplification. Like terms are terms that contain the same variables raised to the same power. In our example, after distributing, we ended up with the expression \-4 - 5t + 65\.

Here, we can only combine the constant terms \(-4\ and \65\), because they do not have any variables. So, combining these gives us: \-4 + 65 = 61\.

It's essential to note that the term \-5t\ is left unchanged since there are no other \ t\ terms to combine it with. This step helps to simplify our expression by reducing the number of terms.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its most concise form. This process often includes using the distributive property and combining like terms.

In our exercise, the initial expression was \(-4-5(t-13)\). After distributing and combining like terms, we refined it to \-5t + 61\.

Simplifying makes the expression easier to understand and manage, especially for solving equations or further mathematical operations.

Let's recap the steps:

• Distribute \(-5\) to each term inside the parentheses: \(-5(t-13) = -5t + 65\)
• Combine the constant terms: \(-4 + 65 = 61\)
• Write the final simplified expression: \-5t + 61\.

Result: The simplified expression is \-5t + 61\.