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Chapter 7: Problem 15

\(\frac{y+5}{y-2} \cdot(2 y)\)

Short Answer

Expert verified
\(\frac{2y^2+10y}{y-2}\)

Step by step solution

01

Simplify the Fraction

First, perform the operation that is in the fraction without the multiplication. Here, there is nothing to simplify in \(\frac{y+5}{y-2}\), so we carry it down directly to the next step.
02

Apply the distributive property

Next, multiply the fraction \(\frac{y+5}{y-2}\) by (2y). Multiply each term in the numerator of the fraction by (2y) using the distributive property to get \(\frac{2y(y+5)}{y-2}\), which simplifies to \(\frac{2y^2+10y}{y-2}\).
03

Provide the Final Answer

To finalize, write down the final answer: \(\frac{2y^2+10y}{y-2}\) ensuring there are no further operations to perform on.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Distributive Property
The distributive property is a fundamental concept in arithmetic and algebra that allows you to simplify expressions. It's like distributing something evenly across a group. In our exercise, we use this property to multiply a fraction by a polynomial.

The distributive property states that for any numbers or expressions, \[ a(b + c) = ab + ac \]. This means you multiply the term outside the parenthesis by each term inside.

In the problem, we have \( \frac{y+5}{y-2} \cdot (2y) \). Using the distributive property, you multiply \( 2y \) across each term in the numerator:
  • First, multiply \( 2y \times y = 2y^2 \).
  • Then, \( 2y \times 5 = 10y \).
This gives us \( \frac{2y^2 + 10y}{y-2} \), which is a critical step in solving our expression.
The Art of Simplifying Fractions
Simplifying fractions involves making an expression as straightforward as possible. In the exercise, the fraction originally given \( \frac{y+5}{y-2} \) is already in a form that can't be simplified further, since \( y+5 \) and \( y-2 \) share no common factors other than 1.

However, after applying the distributive property, our fraction becomes \( \frac{2y^2 + 10y}{y-2} \).If the numerator and the denominator shared common factors, we'd factor them out to simplify the expression. But in this case, there are no common factors between \( 2y^2 + 10y \) and \( y-2 \).Therefore, our fraction is already simplified.
Exploring Elementary Algebra Concepts
Elementary algebra focuses on the manipulation and solving of equations and expressions using symbols. It provides foundational skills for further study in mathematics.

In our example, algebraic skills are applied to multiply terms and simplify expressions. The starting point is to recognize how expressions like \( \frac{y+5}{y-2} \) interact with multiplication, specifically using properties like the distributive property.
  • We use symbols like \( y \) to represent numbers, making it easier to work with general cases instead of specific values.
  • Recognizing when and how to apply algebraic rules helps to manipulate expressions efficiently.
By mastering simple algebraic operations, students gain the tools to move on to more complex topics in mathematics.

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Most popular questions from this chapter

Precision Describe how to rewrite a complex fraction as a product.

\(\frac{10 x}{(x-3)^{2}}-\frac{7 x}{(x-3)}\)

\(\frac{6 x}{x-11}+1=\frac{3}{x-11}\)

Writing Can you assume that two people will complete a task twice as quickly as one of the two people working alone? Explain.

\(\frac{x}{x-4}-\frac{9}{x}=\frac{3}{x-4}\)
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