Problem 111 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 36\left(\frac{2}{9} x-\frac{3}{4}\right)+36\left(\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 8x - 9 \).

Step 1: Distribute the 36

To simplify the expression, start by distributing the 36 across the terms inside the parentheses. Apply distribution to each term separately: \[ 36 \left( \frac{2}{9}x \right) - 36 \left( \frac{3}{4} \right) + 36 \left( \frac{1}{2} \right) \].This results in:\[ \frac{72}{9}x - \frac{108}{4} + \frac{36}{2} \].

Step 2: Simplify Fractions

Now, simplify each fraction:- \( \frac{72}{9} = 8 \), so the term becomes \( 8x \).- \( \frac{108}{4} = 27 \), so the term becomes \( 27 \).- \( \frac{36}{2} = 18 \), so the term becomes \( 18 \).The expression is now \( 8x - 27 + 18 \).

Step 3: Combine Like Terms

Combine the constant terms in the expression to simplify further:\[ 8x - 27 + 18 = 8x - 9 \].Now the expression cannot be simplified further.

Key Concepts

Distribution Property

The distribution property, also known as the distributive law, allows us to multiply a single term by each term inside a set of parentheses. Imagine you have a gift bag with smaller bags inside, and you want to give each small bag the same number of treats. The distribution property lets you ensure that each small bag gets its equal share.

In mathematics, it works the same way. When you have an expression such as \[ 36(\frac{2}{9}x - \frac{3}{4}) + 36(\frac{1}{2}) \], you "distribute" 36 to everything inside the parentheses. This means multiplying 36 with \( \frac{2}{9}x \) and then with \( \frac{3}{4} \), and likewise, multiplying 36 by \( \frac{1}{2} \) separately.

When applied, it results in the expression: \[ \frac{72}{9}x - \frac{108}{4} + \frac{36}{2} \]. This step sets the stage for simplification in algebraic expressions.

Simplifying Fractions

Once distribution is complete, the next step involves simplifying the fractions that arise. Simplifying fractions means finding an equivalent fraction where the numerator and the denominator have no common factors other than 1.

For example, in the expression\[ \frac{72}{9}x - \frac{108}{4} + \frac{36}{2} \], each fraction can be simplified as follows:

• \( \frac{72}{9}x \) simplifies to \( 8x \) because 72 divided by 9 equals 8.
• \( \frac{108}{4} \) simplifies to 27, since 108 divided by 4 is 27.
• \( \frac{36}{2} \) simplifies to 18, as 36 divided by 2 is 18.

Simplifying fractions reduces complexity and makes the entire expression easier to work with.  By ensuring each part of the expression is as simple as possible, you are better prepared to **combine like terms**, which forms the next part of your algebraic journey.

Combining Like Terms

Combining like terms is a vital process to ease and simplify algebraic expressions. Think of like terms as tokens that belong to the same category, such as fruits in a basket. You wouldn't combine apples and oranges, but you can combine different apples and consolidate them into one neat group.

In the expression \[ 8x - 27 + 18 \], the term \( 8x \) stands alone as the only term involving \( x \). However, the numbers \( -27 \) and \( 18 \) are constant terms and can be combined.

Adding these like terms simply means performing the arithmetic operation:

• -27 plus 18 equals -9.

As a result, the entire expression becomes\[ 8x - 9 \]. Knowing how to combine like terms correctly helps streamline calculations and reveals the expression's simplest form.

This technique is especially useful in solving algebraic equations and providing clarity to more complex mathematical problems.