Problem 71 - Algebra A Combined Function

Question

Identify each statement as true or false. If false, correct the multiplication. See the Concept Check in this section. $$ \frac{x}{5} \cdot \frac{x+3}{4}=\frac{2 x+3}{20} $$

Step-by-Step Solution

Answer

False; the correct multiplication is \( \frac{x^2 + 3x}{20} \).

Step 1: Expand the Left Side of the Equation

The given expression is \( \frac{x}{5} \cdot \frac{x+3}{4} \). Multiply the numerators and denominators: \[ \frac{x \cdot (x+3)}{5 \cdot 4} = \frac{x^2 + 3x}{20} \]

Step 2: Compare Both Sides of the Equation

Now, compare the expanded form \( \frac{x^2 + 3x}{20} \) from Step 1 with the right side of the original equation \( \frac{2x + 3}{20} \).

Step 3: Determine the Truth of the Statement

Since \( x^2 + 3x \) does not equal \( 2x + 3 \), the statement is false.

Step 4: Correct the Multiplication

The correct multiplication is already provided in Step 1: \( \frac{x}{5} \cdot \frac{x+3}{4} = \frac{x^2 + 3x}{20} \).

Key Concepts

Incorrect Equations

Equations can sometimes be misleading, especially when they combine variables and operations like multiplication. In our exercise, the purpose was to check if the equation involving fractions was correct. Often, mathematical errors occur when equations are not properly set up or simplified.
This is why it’s crucial to always expand and compare both sides carefully.

• Verify each step: Ensure you go through all steps slowly to avoid oversight.
• Recheck calculations: Small mistakes in arithmetic can lead to incorrect conclusions.

The process of identifying incorrect equations involves a detailed expansion of terms and simplification. If both sides do not equate after simplification, then it signals a potential error in the expression or a misunderstanding of the equation structure.

Algebraic Expressions

Algebraic expressions involve combinations of numbers, variables, and operations. They allow us to represent general numbers and form the basis for constructing equations. In this exercise, we used expressions like \( \frac{x}{5} \cdot \frac{x+3}{4} \). Expanding these expressions helps in understanding how each variable interacts within an equation.
When dealing with algebraic terms:

• Understand each term: Variables like \( x \) can represent any number, which means they need careful manipulation.
• Pay attention to operators: Addition, subtraction, multiplication, and division have different impacts on equation outcomes.

Mastery of algebraic expressions equips you with the tools to break down complex equations into simpler parts. This makes solving them a manageable task, as seen in the expanded form in our exercise.

Multiplication of Fractions

Multiplying fractions is a key operation in math, where you multiply the numerators together and the denominators together. In algebra, this operation applies to variable expressions as well.
In our example, multiplying \( \frac{x}{5} \) by \( \frac{x+3}{4} \) required:

• Multiplying numerators: \( x \cdot (x+3) = x^2 + 3x \).
• Multiplying denominators: \( 5 \cdot 4 = 20 \).

The result \( \frac{x^2 + 3x}{20} \) highlighted the importance of correctly handling each fraction part.
Remember:

• Always simplify: After multiplication, check if the fraction can be reduced.
• Check for common factors: They may simplify your expression significantly.

Understanding these steps ensures that the multiplication of fractions, whether numerical or algebraic, is performed accurately and provides correct results.