Problem 376 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 9(5 u+8)+2(u-6) $$

Step-by-Step Solution

Answer

47u + 60

Step 1 - Apply the Distributive Property to the first term

Distribute 9 to both terms inside the parentheses: \[9(5u) + 9(8) = 45u + 72\]

Step 2 - Apply the Distributive Property to the second term

Distribute 2 to both terms inside the parentheses: \[2(u) + 2(-6) = 2u - 12\]

Step 3 - Combine like terms

Add the results from step 1 and step 2 together: \[45u + 72 + 2u - 12\]

Step 4 - Simplify the expression

Combine the like terms for the final answer: \[45u + 2u + 72 - 12 = 47u + 60\]

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that helps to make mathematical expressions as concise as possible. In this exercise, we started with the expression \[9(5u + 8) + 2(u - 6)\].
The goal here is to get an expression that is easy to understand and work with.
We simplify expressions by using various algebraic techniques like the Distributive Property.
By applying the Distributive Property, we multiply the number outside the parentheses with each term inside the parentheses. This step transforms the original expression into an easier one to handle.

For instance:

• We distribute 9 through \[5u + 8\]
• Then, we distribute 2 through \[u - 6\]

The exercise's first step shows this: \[9(5u) + 9(8) = 45u + 72\]
After distributing, the terms are easy to combine.

Combining Like Terms

Combining like terms is another essential part of simplifying algebraic expressions.
Terms are 'like' if they have the same variables raised to the same powers. In the exercise, we have terms with the variable 'u'.

Coming from the distribution steps, we gathered:

• \[45u\]
• \[72\]
• \[2u\]
• \[-12\]

To combine like terms means to add or subtract the coefficients of these terms.
The solution adds \[45u\] and \[2u\] to get \[47u\]. Similarly, we combine constants \[72 - 12 = 60\].

So, combining like terms yields: \[45u + 72 + 2u - 12 = 47u + 60\]

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations or expressions to solve for variables or simplify the problem.
It’s essential for solving complex algebraic equations step-by-step.

The last part of the exercise uses algebraic manipulation techniques.
We started with the expression \[9(5u + 8) + 2(u - 6)\], applied the Distributive Property, and then combined like terms.

The steps are as follows:

• Distribute constants into parentheses.
• Rearrange and combine like terms.
• Simplify to get the final expression.

This problem highlights how algebraic concepts work together:
\[9(5u + 8) + 2(u - 6) \rightarrow 45u + 72 + 2u - 12 \rightarrow 47u + 60\]
Using these processes, algebraic manipulation transforms complex expressions into manageable ones.