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Chapter 1: Problem 54

In Problems \(52-55,\) decide whether the expressions can be put in the form $$ \frac{a x}{a+x} . $$ For those that are of this form, identify \(a\) and \(x\). $$ \frac{8 y}{4 y+2} $$

Short Answer

Expert verified
Answer: Yes, the given expression can be put in the form \(\frac{a x}{a+x}\) with \(a = 4\) and \(x = y\).

Step by step solution

01

Identify the given expression

The given expression is \(\frac{8y}{4y+2}\). We need to see if this expression can take the form \(\frac{a x}{a+x}\).
02

Simplify the expression

In order to simplify the expression, we can factor out the common factor in numerator and denominator. In this case, both numerator and denominator have a common factor of \(2\). So, we can divide both by \(2\): $$ \frac{8y}{4y+2} = \frac{2(4y)}{2(2y+1)} = \frac{4y}{2y+1} $$
03

Compare the simplified expression with the desired form

Now, let's compare the simplified expression \(\frac{4y}{2y+1}\) with the desired form \(\frac{a x}{a+x}\). To do that, we can assign values to \(a\) and \(x\) using the simplified expression: $$ a = 4, \qquad x = y $$ By using these values, we can rewrite the desired form: $$ \frac{a x}{a+x} = \frac{4y}{4+y} $$
04

Conclusion

We have successfully transformed the given expression \(\frac{8y}{4y+2}\) into the desired form \(\frac{a x}{a+x}\). The values for \(a\) and \(x\) are: $$ a = 4, \qquad x = y $$ Thus, the given expression can be put in the form \(\frac{a x}{a+x}\) with \(a = 4\) and \(x = y\).

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Key Concepts

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

Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. In simpler terms, they are similar to fractions but instead of just numbers, they consist of expressions with variables. When dealing with rational expressions, it鈥檚 important to understand that they behave much like regular fractions.
  • Both the numerator and the denominator must be polynomial expressions. For example, \( \frac{8y}{4y+2} \), where both the top and bottom are polynomial expressions.
  • We need to consider the domain, which excludes values making the denominator zero.
  • Simplifying rational expressions often helps in solving or rewriting them into a desired form.
Understanding the properties of rational expressions is crucial as it enables us to perform operations like addition, subtraction, multiplication, and division, much like with fractions!
Expression Simplification
Simplifying expressions is an essential skill in algebra. This involves reducing the expression to its simplest form, making it easier to handle.
To simplify rational expressions:
  • Look for common factors in the numerator and the denominator.
  • Factor them out whenever possible, as this can lead to cancellation of common terms.
In our exercise,
- We started with \( \frac{8y}{4y+2} \).
- The next step was factoring out the greatest common factor, which is \( 2 \).
- This provides \( \frac{2(4y)}{2(2y+1)} \), allowing us to simplify further to \( \frac{4y}{2y+1} \).Thus, simplifying involves breaking down complex expressions into their simplest form, not only for easier computation but also to reveal the underlying structure of the expression.
Factoring in Algebra
Factoring is a critical concept in algebra that involves expressing a polynomial as a product of simpler polynomials. It is a primary tool for simplifying expressions and solving equations.
To factor an expression:
  • Identify any common factors of the terms.
  • Rewrite the expression to show these factors.
For instance, to simplify \( \frac{8y}{4y+2} \), we first factor the greatest common factor of the terms in the numerator and denominator, which is \( 2 \).
This leads us to a simpler form \( \frac{4y}{2y+1} \).
Factoring is not only limited to simplifying expressions but is also crucial for solving quadratic equations and other algebraic problems by setting each factor to zero to find the roots.

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Most popular questions from this chapter

In Problems \(43-45\) assume a person originally owes \(\$ B\) and has made \(n\) payments of \(\$ p\) each. Assume no interest is charged. What does the expression \(B-n p\) represent in terms of the person's debt?

In Problems \(63-64\), a farm planted with \(A\) acres of corn yields \(b\) bushels per acre. Each bushel brings \(\$ p\) at market, and it costs \(\$ c\) to plant and harvest an acre of corn. What do the expressions \(c A\) and \(p b\) represent for the farm?

In Exercises \(20-23,\) write an expression for the sequence of operations. $$ \text { Add } 3 \text { to } x \text { , double, subtract } 1 \text { from the result. } $$

The number of people who attend a concert is \(160-p\) when the price of a ticket is \(\$ p\). (a) What is the practical interpretation of the \(160 ?\) (b) Why is it reasonable that the \(p\) term has a negative sign? (c) The number of people who attend a movie at ticket price \(\$ p\) is \(175-p\). If tickets are the same price, does the concert or the movie draw the larger audience? The number of people who attend a concert is \(160-p\) when the price of a ticket is \(\$ p\). (a) What is the practical interpretation of the \(160 ?\) (b) Why is it reasonable that the \(p\) term has a negative sign? (c) The number of people who attend a movie at ticket price \(\$ p\) is \(175-p .\) If tickets are the same price, does the concert or the movie draw the larger audience?

Does the equation have a solution? Ex plain how you know without solving it. $$ \frac{x+3}{5+x}=1 $$
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