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Chapter 4: Problem 31

Simplify.$$\frac{x y+6}{3 y}$$

Short Answer

Expert verified
The simplified version of \( \frac{x y+6}{3 y} \) is itself as the expression cannot be simplified further.

Step by step solution

01

Identify and simplify numerator and denominator

Firstly, identify the numerator \(x y+6\) and the denominator \(3 y\). We can simplify the numerator \(x y+6\) by factorizing it, but in this case, it can't be factorized further. Also, the denominator \(3 y\) is already in its simplest form.
02

Cancel out common terms

Now, try to find out if there are any common terms in the numerator and the denominator that could be canceled out. However, in this case there are no common terms that could be canceled out. Therefore, the expression stays the same.
03

Final expression

So, the final simplified version of the given expression \(\frac{x y+6}{3 y}\) is itself, as no further simplification can be done.

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Key Concepts

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

Simplification
Simplification in algebra is like tidying up a messy room. It means making an expression look as neat and simple as possible, without changing its value. For algebraic expressions, simplification often involves reducing expressions by canceling terms or finding common factors. Sometimes, we combine like terms, resolve arithmetic operations, and apply basic algebraic identities.
  • Simplifying fractions is about making sure they are in their simplest form.
  • Look for common factors between the numerator and the denominator.
  • Remove or "cancel" those common factors to simplify the expression.
In the given exercise, simplification involved checking both, the numerator and the denominator, for any shared factors we could cancel, which unfortunately was not possible here. Hence, the expression stayed unchanged.
Numerator and Denominator
Understanding numerators and denominators is crucial for managing fractions. These are the top and bottom parts of a fraction, respectively. The numerator indicates how many parts we have, while the denominator tells us about the total number of equal parts something is divided into.
  • The numerator is the expression above the fraction line, e.g., in \(\frac{xy+6}{3y}\), \(xy+6\) is the numerator.
  • The denominator is below the fraction line, indicating the total or whole, like \(3y\) in \(\frac{xy+6}{3y}\).
To simplify fractions, it's important to analyze both these parts separately, check if they share any common factors, and then cancel those out to reduce the fraction to its simplest form.
Factorization
Factorization is essential in simplifying algebraic expressions. It involves breaking down an expression into a product of its factors. These factors, when multiplied together, give back the original expression. Factorization can often reveal opportunities for simplification that are not immediately visible.
  • Factorization helps in identifying common factors in the numerator and the denominator within fractions.
  • If we successfully factorize the numerator or denominator, we might find common terms to cancel out.
  • However, sometimes an expression may not be further factorized, as was the case in \(xy+6\).
Mastering factorization helps to simplify complex expressions effectively and is a powerful tool in your algebraic toolbox.

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Most popular questions from this chapter

Writing the expression \((x+y)^{2}\) as \(x^{2}+y^{2}\) illustrates a common error. Explain.

Perform the operations. $$\begin{array}{r}-6 x^{3} z-4 x^{2} z^{2}+7 z^{3} \\\\+\quad-7 x^{3} z+9 x^{2} z^{2}-21 z^{3}\end{array}$$

Simplify or solve as appropriate. $$(b+2)(b-2)+2 b(b+1)$$

Find a polynomial whose value will be 1 if you substitute \(\frac{3}{2}\) for \(x\).

Simplify. $$7\left(x^{2}+3 x+1\right)+9\left(x^{2}+3 x+1\right)-5\left(x^{2}+3 x+1\right)$$
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