Problem 80 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-3(2 t+4)+8(2 t-4)\)

Step-by-Step Solution

Answer

10t - 44

Step 1 - Distribute the first term

Apply the distributive property to \(-3(2t + 4)\). This means multiplying -3 by both 2t and 4:\(-3 * 2t + (-3) * 4 = -6t - 12\).

Step 2 - Distribute the second term

Next, apply the distributive property to \(8(2t - 4)\). This involves multiplying 8 by both 2t and -4:\(8 * 2t + 8 * -4 = 16t - 32\).

Step 3 - Combine like terms

Add together the expressions from steps 1 and 2: \(-6t - 12 + 16t - 32\). To combine like terms, group the terms with \t\ and the constant terms:\(-6t + 16t - 12 - 32\).

Step 4 - Simplify the combined expression

Combine the like terms to simplify the expression: \(-6t + 16t = 10t\) and \-12 - 32 = -44\. So the simplified expression is: \(10t - 44\).

Key Concepts

distributive property

The distributive property is a fundamental concept in algebra. It allows us to simplify expressions by multiplying each term inside a parenthesis by a factor outside the parenthesis. In this exercise, we used the distributive property to break down the expression \(-3(2t+4)\) and \(8(2t-4)\). Here’s how it works:

• For \(-3(2t+4)\), we distribute -3 to both \(2t\) and 4: \(-3 \times 2t + (-3) \times 4 = -6t - 12\).
• Similarly, for \(8(2t-4)\), we distribute 8 to both \(2t\) and -4: \(8 \times 2t + 8 \times -4 = 16t - 32\).

This step is crucial for breaking down more complicated expressions into manageable parts.

combining like terms

Combining like terms is another important technique when simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In the expression from our exercise, we have \(-6t-12+16t-32\):

• Like terms with \(t\) are \(-6t\) and \(16t\).
• Constant terms are \(-12\) and \(-32\).

To combine like terms, we group them together: \(-6t+16t\) and \(-12-32\). This makes it easier to simplify the expression in the next step.

simplification

The final step in the process is simplification. Here, we combine the grouped terms to get a simpler expression:

• First, we add the \(t\) terms: \(-6t+16t = 10t\).
• Next, we add the constant terms: \(-12-32 = -44\).

By combining these, the fully simplified expression is \(10t-44\). Each step builds upon the previous steps, making simplification a more manageable and understandable process.