Problem 94 - Intermediate Algebra

Question

Simplify each expression. $$2 t z+5(t z-4)-10(8-t z)$$

Step-by-Step Solution

Answer

The expression simplifies to \(17tz - 100\).

Step 1: Expand the Expression

Start by expanding the equation using distributive property. 1. Expand the expression inside the parentheses: \(2tz + 5(tz - 4) - 10(8 - tz)\)This becomes: \(2tz + 5tz - 20 - 80 + 10tz\)

Step 2: Combine Like Terms

Now, combine the like terms in the expanded expression:1. Add the \(tz\) terms: \(2tz + 5tz + 10tz = 17tz\)2. Combine the constants: \(-20 - 80 = -100\)The expression thus simplifies to:\(17tz - 100\).

Key Concepts

Distributive Property

The distributive property is a very useful tool in algebra. It helps when you need to expand expressions and remove parentheses. In simple terms, when you have a term outside a set of parentheses, you multiply it by each term inside. This is how you "distribute" the multiplication across the terms within the parentheses.

In math terms, the rule is written as:

• If you have a(b + c), it becomes ab + ac.
• Similarly, a(b - c) becomes ab - ac.

For our original exercise, we applied the distributive property twice:

• First, expand 5(tz - 4) to get 5tz - 20.
• Then, expand -10(8 - tz) to get -80 + 10tz.

By using this property, we could effectively remove the parentheses and start combining terms. Understanding this property is foundational because it allows you to work with and simplify more complex expressions efficiently.

Combining Like Terms

Combining like terms is another essential skill in simplifying expressions. Like terms are terms that have identical variable parts. In our example, this meant looking for terms that have the variable combination 'tz'.

Once we expanded our expression using the distributive property, we had three sets of 'tz' terms: 2tz, 5tz, and 10tz.

• These aren't just random terms — they relate directly to one another because they share the same variables.
• We combine them by adding their coefficients, resulting in 17tz (2 + 5 + 10 = 17).

After combining the 'tz' terms, don't forget about constant numbers. In our case, the terms -20 and -80 were combined to get -100.

This step makes expressions simpler and easier to read. Recognizing and combining like terms will also make solving equations and other algebraic problems more straightforward.

Simplifying Expressions

Simplifying expressions is a crucial part of algebra that helps make complex equations easier to manage. It involves reducing an expression to its simplest form, which is easier to understand and work with.

With the use of distributive property and combining like terms, we managed to take the originally complex expression, 2tz + 5(tz - 4) - 10(8 - tz), and simplify it into a cleaner expression: 17tz - 100.

• Simplifying helps prevent mistakes, especially when more complicated operations or solutions are required.
• It also makes subsequent steps, like solving equations or graphing functions, clearer and more efficient.

By practicing these techniques, simplifying expressions becomes second nature, allowing for greater focus on higher-level problem-solving. Remember, simplification is not just a tool — it's an essential aspect of effective algebraic work.