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Chapter 4: Problem 30

Simplify $$\frac{8 x+12 y}{4 x y}$$

Short Answer

Expert verified
The simplified form of the given expression \(\frac{8x + 12y}{4xy}\) is \(\frac{2x + 3y}{xy}\).

Step by step solution

01

Identify Common Factor

Look at the numerator and identify if there are numbers that can divide both terms. In the case of \(8x + 12y\), the common factor is 4.
02

Apply the Factoring Principle

Factor out the common number from both numerator and denominator. This yields \(\frac{4(2x + 3y)}{4xy}\).
03

Eliminate Common Factors

Eliminate the common factor, 4, on both the numerator and the denominator. This yields \(\frac{2x + 3y}{xy}\).

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Key Concepts

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

Factoring
Factoring in algebra involves breaking down expressions into simpler components. This process can help simplify or solve equations more easily.
To factor an expression, you search for values that can multiply to form each term within the expression.

For example, with the expression in our numerator, \(8x + 12y\), the first step is to identify values that can multiply together to give \(8x\) and another set for \(12y\).
  • The number 4 is common in both terms because \(8x = 4 imes 2x\) and \(12y = 4 imes 3y\).
By identifying and applying this shared factor, you can rewrite the expression.
This step of factoring sets up the simplification process by providing terms that can be easily canceled.
Common Factor
A common factor is a number or expression shared by two or more terms.
Recognizing common factors is crucial in algebra, as it helps in simplifying expressions.

In the expression \(8x + 12y\), we identify the number 4 as it divides both terms equally:
  • Since 4 is a divisor of both 8 and 12, it qualifies as a common factor.
When common factors are identified and factored out, expressions become neater and easier to manage.
Isolating the common factor, as was done in this exercise, aids in revealing simplified forms of the expressions.
Numerator and Denominator Reduction
Once factored, the next step is to reduce the fraction by eliminating common factors.
Reduction involves dividing out the common components from both the numerator and the denominator.

In the expression \(\frac{4(2x + 3y)}{4xy}\), the 4 in the numerator and the 4 in the denominator are common.
  • By dividing both by 4, we simplify the expression to \(\frac{2x + 3y}{xy}\).
This process of canceling simplifies the entire fraction.
By removing common components, you make the expression easier to interpret and manage. Reducing fractions is a fundamental aspect of algebra that streamlines calculations and solutions.
Algebraic Expression Simplification
Simplifying algebraic expressions involves making them as concise as possible without changing their value.
It's a blend of factoring, recognizing common factors, and canceling out unnecessary components.

The initial expression \(\frac{8x + 12y}{4xy}\) might look complex, but using simplification techniques leads us to a much simpler form.
  • First, by factoring out common elements like the number 4.
  • Second, by reducing the fraction by canceling common factors across the numerator and denominator.
The final simplified expression \(\frac{2x + 3y}{xy}\) is not only easier to handle but serves the same mathematical purpose as the original.
Mastering these techniques helps students transition from basic calculations to more advanced problem-solving skills.

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