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Chapter 7: Problem 71

Simplify. $$ \frac{1-\frac{z}{2+2 z}-2 z}{\frac{2 z}{5 z-2}-3} $$

Short Answer

Expert verified
The simplified form is: \frac{(2 - 2z)(5z - 2)}{(2+2z)(-13z + 6)}

Step by step solution

01

Simplify the numerator

First, simplify the numerator: \frac{1-\frac{z}{2+2z}-2z}. To do this, find a common denominator for the terms in the numerator.The common denominator for the terms is (2+2z). So, we have: \[ 1 = \frac{2+2z}{2+2z} \]Then, it becomes: \[ 1-\frac{z}{2+2z}-2z = \frac{2+2z-z-4z(2+2z)}{2+2z} \ = \frac{2+2z-z-4z}{2+2z} \ = \frac{2+2z-z-4z}{2+2z} \ = \frac{2-2z}{2+2z} \]
02

Simplify the denominator

Next, simplify the denominator: \frac{2 z}{5 z-2}-3. Find the common denominator for the terms in the denominator, which is (5z-2):\[ \frac{2z}{5z-2} - 3 = \frac{2z - 3(5z-2)}{5z-2} \] Simplify further: \[ \frac{2z - 15z + 6}{5z-2} = \frac{-13z + 6}{5z-2} \]
03

Divide the simplified numerator by the simplified denominator

Finally, divide the simplified numerator by the simplified denominator:\[ \frac{\frac{2-2z}{2+2z}}{\frac{-13z + 6}{5z-2}} \] This is equivalent to multiplying by the reciprocal:\[= \frac{2-2z}{2+2z} * \frac{5z-2}{-13z + 6} \] Therefore:\[= \frac{(2 - 2z)(5z - 2)}{(2+2z)(-13z + 6)} \]
04

Simplify further if possible

Investigate potential factorization in the equation to simplify it further if possible. In this case, no common factors exist. Therefore, the simplified form remains:\[ \frac{(2 - 2z)(5z - 2)}{(2+2z)(-13z + 6)} \]

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Key Concepts

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

Common Denominator
To simplify algebraic fractions, finding a common denominator is crucial. This helps combine fractions by converting them into a common base.
Let's break it down:
When you have fractions like \(\frac{z}{2 + 2z}\), identify the denominators. Here it is \(2 + 2z\).
  • Rewrite other terms using the common denominator: Since \(1\) is part of the mix, express it differently: \(\frac{2 + 2z}{2 + 2z} = 1\).
  • Combine fractions: Now, subtract fractions and navigate algebraic operations, like this: \(\frac{2 + 2z}{2 + 2z} - \frac{z}{2 + 2z} - 2z\).

After finding and using the common denominator, the expression becomes manageable. Accurately applying this concept significantly simplifies the problem.
Factorization
Factorization breaks down complex algebraic expressions into simpler ones by identifying common factors.
Consider the given problem:
  • The numerator is \(\frac{2 - 2z}{2 + 2z}\).
  • Simplifying further, identify common factors. Notice \(2\) as a common factor: \(\frac{2(1 - z)}{2(1 + z)}\).

This helps simplify the expression by canceling out the common factor 2: thus, \(\frac{1 - z}{1 + z}\).
In more complex scenarios, factorization is a helpful tool. Break down the parts and simplify using common terms.
Reciprocal
The reciprocal of a fraction is swapping its numerator and denominator. This concept is essential, especially while dividing fractions.
For example, dividing by \(\frac{-13z + 6}{5z - 2}\) is the same as multiplying by its reciprocal. \(\frac{5z - 2}{-13z + 6}\):
  • Given: \(\frac{2 - 2z}{2 + 2z} \/ \frac{-13z + 6}{5z - 2}\).
  • Find the reciprocal: \(\frac{5z - 2}{-13z + 6}\).
  • Multiply: \(\frac{2 - 2z}{2 + 2z} * \frac{5z - 2}{-13z + 6}\).

Reciprocals simplify complicated fraction operations. Always swap and multiply.

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Most popular questions from this chapter

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