Problem 33 - Elementary Algebra

Question

\(6(3 u)=(6 \cdot 3) u\)

Step-by-Step Solution

Answer

Yes, \( 6(3u) = (6 \cdot 3)u \) simplifies to \(18u = 18u\).

Step 1: Identify the Expression

Notice the given expression: \[ 6(3u) = (6 \cdot 3)u \]We need to verify whether both sides of the equation are equal.

Step 2: Simplify Inside the Parentheses

Simplify the term inside the parentheses on the left-hand side: \[ 6(3u) = 6 \cdot 3 \cdot u \]

Step 3: Perform the Multiplication

Multiply the constants on the left-hand side: \[ 6 \cdot 3 \cdot u = 18u \]

Step 4: Compare Both Sides

Now, compare the simplified left-hand side to the right-hand side: \[ 18u = 18u \]Both sides are equal, confirming that the original equation is true.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. In our exercise, we have the expression \(6(3u) = (6 \cdot 3)u\). Here, \(3u\) is the algebraic part, with \(3\) being the coefficient and \(u\) the variable. Algebraic expressions can be manipulated in various ways using algebraic properties like the Distributive Property. The Distributive Property helps in breaking down expressions to make simplification easier. Learning to work with these expressions is a crucial skill in algebra and mathematics overall. Understanding how to recognize and manipulate algebraic components will greatly assist in solving more complex problems as you progress.

Simplification

Simplification is the process of reducing an algebraic expression to its simplest form. In the exercise, we start with simplifying \(6(3u)\). To do this, we look inside the parentheses and see that \(3u\) can be separated. Applying the Distributive Property, \(6(3u)\) becomes \(6 \cdot 3 \cdot u\). This step is critical because it sets up the expression for easier multiplication and further simplification. Simplification usually involves:

• Combining like terms
• Using distributive property
• Reducing fractions, if any

In our example, once you simplify inside the parentheses, you can move on to the next step, which involves multiplication.

Multiplication

Multiplication in algebra follows the same basic principles as arithmetic multiplication but often involves variables. In our exercise, once we have broken down \(6(3u)\) to \(6 \cdot 3 \cdot u\), we perform the multiplication of the constants first. So, \(6 \cdot 3 = 18\), and then we multiply by the variable \(u\), which gives us \(18u\). This ability to distribute and multiply is essential for simplifying and solving algebraic expressions and equations. It's important to follow the order of operations (sometimes remembered as PEMDAS) to ensure calculations are done correctly. By practicing these steps, you can solve similar problems with ease.