Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(5 l \div 3 l \times(9-6)\)

Step-by-Step Solution

Answer

The expression simplifies to 5.

Step 1: Simplify the expression inside the parentheses

First, let's handle the expression inside the parentheses: \[9 - 6 = 3\]

Step 2: Simplify the division

Next, simplify the division operation in the expression. We have:\[5l \div 3l\]Since there are no constants or variables that cancel out entirely, we simply cancel the common variable \(l\):\[\frac{5l}{3l} = \frac{5}{3}\]

Step 3: Multiply the result by the simplified term from Step 1

Finally, multiply the result from Step 2 by the simplified expression from Step 1. We have:\[\frac{5}{3} \times 3 = \frac{5 \times 3}{3}\]The \(3\)s in the numerator and denominator cancel out, leaving:\[5\]

Key Concepts

Division and Multiplication

In algebra, understanding how to handle division and multiplication in expressions with variables is crucial. These operations are performed following the order of operations, often abbreviated as PEMDAS (which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
When you're working with an expression like \(5l \div 3l \times (9-6)\), you first need to focus on the division and multiplication steps. In algebra, these operators are equally prioritized and resolved from left to right as they appear in the expression.

Here's a quick breakdown:

• Division: This means you'll divide one expression by another, such as \(\frac{5l}{3l}\). Because the variable \(l\) is present in both the numerator and denominator, it can be cancelled out, simplifying to \(\frac{5}{3}\).
• Multiplication: After division, you'll multiply this result by another number or expression. In the example, after simplifying the division, you multiply \(\frac{5}{3}\) by the result of \(9-6\), which is 3, giving \(\frac{5 \times 3}{3}\).
  

Understanding these operations helps in simplifying complex expressions efficiently, keeping calculations accurate and simplifying steps along the way.

Parentheses in Algebra

Parentheses play a pivotal role in algebra, as they indicate operations that must be performed first according to the order of operations. This grouping feature allows you to simplify expressions by isolating terms to reduce possible errors in calculation.
In our example, the expression \((9-6)\) is enclosed in parentheses, signaling that it should be simplified before any multiplication or division.

Let's look at how this works:

• Simplifying Parentheses: Calculate the expression inside the parentheses first, which in this case is \((9-6)\). Simplifying it gives \(3\).
• Application in Expressions: Once simplified, replace the original parenthetical expression with its simplified result before proceeding with other operations like multiplication or division.

In algebraic expressions, always resolving what's inside parentheses first ensures correctness and clarity in solving equations. It helps in neatly progressing through problem-solving steps while avoiding mistakes.

Variables in Algebra

Variables are symbols used to represent unknown values in algebra. They are often denoted by letters like \(x, y, z\), or in this case, \(l\). In algebraic expressions, understanding how to manage these variables is key to simplifying and solving equations.
The division of expressions containing variables, such as \(5l \div 3l\), involves cancelling variables where possible.

Here's how it works:

• Cancelling Variables: With \(5l \div 3l\), both the numerator and the denominator have the variable \(l\). They cancel each other out because any number (or variable) divided by itself equals 1, giving a simplified ratio of \(\frac{5}{3}\).
• Significance: Canceling common variables reduces complexity, making expressions easier to interpret and solve. It also highlights the numerical relationships between terms.

Algebra often relies on manipulating variables to discover solutions or simplify expressions. Mastering these techniques leads to greater proficiency and confidence in solving algebraic problems.