Chapter 11: Problem 13
Simplify the expression. $$\frac{2}{3 x-1}-\frac{5 x}{3 x-1}$$
Short Answer
Expert verified
The simplified form of the given expression is \[\frac{-5x + 2}{3x -1}\]
Step by step solution
01
Understand the principle of simplifying fractions
Recognize that for fractions with the same denominator, we can easily subtract or add the numerators. In the expression, both fractions have '3x-1' as their denominator.
02
Apply the principle
Combine the numerators of the fractions by subtraction, as suggested by the exercise. The combined expression would then be \((2 - 5x) / (3x - 1)\).
03
Simplify the numerator
Rewrite the expression \((2 - 5x) / (3x - 1)\) in a simplified form. The simplified form is \((-5x + 2) /(3x-1)\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Subtracting Fractions with Like Denominators
Understanding how to subtract fractions with like denominators is a foundational skill in algebra. The process is straightforward: when two fractions share the same denominator, you can subtract the second numerator from the first without altering the denominator. Picture sharing a pizza between friends—if each slice is the same size (like our common denominator), all you need to worry about when taking slices away is the number of slices (our numerator).
The general formula for subtracting fractions with the same denominator is given by: \[ \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} \]
It's helpful to think of the denominator as the 'shared ground' over which the differences in the numerators play out. To practice this, imagine subtracting the fractions \( \frac{5}{12} \) and \( \frac{3}{12} \); simply take 3 from 5 while keeping the denominator 12, resulting in \( \frac{2}{12} \), which can then be further simplified. The essence of the concept is not in performing complex operations, but in recognizing the simplicity of the fractions' commonality.
The general formula for subtracting fractions with the same denominator is given by: \[ \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} \]
It's helpful to think of the denominator as the 'shared ground' over which the differences in the numerators play out. To practice this, imagine subtracting the fractions \( \frac{5}{12} \) and \( \frac{3}{12} \); simply take 3 from 5 while keeping the denominator 12, resulting in \( \frac{2}{12} \), which can then be further simplified. The essence of the concept is not in performing complex operations, but in recognizing the simplicity of the fractions' commonality.
Combining Numerators Over Common Denominator
Once you've embraced the idea of subtracting fractions with like denominators, the next logical step is combining numerators over a common denominator. It's like coordinating a group project where everyone has their tasks (numerators) and relies on the same resources (the denominator).
To successfully manage this, simply add or subtract the numerators as indicated by the operation while maintaining the denominator as is. For example, if you have \( \frac{7}{4} \) and \( \frac{2}{4} \), and you want to subtract the second from the first, the combined expression is \( \frac{7 - 2}{4} \) resulting in \( \frac{5}{4} \).
The beauty of combining numerators over a common denominator lies in its power to transform a seemingly complex expression into a more manageable single fraction—streamlining the path to further simplification and clarity.
To successfully manage this, simply add or subtract the numerators as indicated by the operation while maintaining the denominator as is. For example, if you have \( \frac{7}{4} \) and \( \frac{2}{4} \), and you want to subtract the second from the first, the combined expression is \( \frac{7 - 2}{4} \) resulting in \( \frac{5}{4} \).
The beauty of combining numerators over a common denominator lies in its power to transform a seemingly complex expression into a more manageable single fraction—streamlining the path to further simplification and clarity.
Simplifying Expressions
Simplifying expressions is much like cleaning up a cluttered room: it makes it easier to understand what's going on and to work within the space. In algebra, this translates to transforming equations into their least complicated form, which often involves combining like terms, distributing multiplications over additions or subtractions, reducing fractions, and eliminating zero terms.
To simplify an expression, always begin by combining like terms. For instance, in \( 3x + 5x \), you can combine the two terms to get \( 8x \). When dealing with fractions, reduce them to their smallest equivalent if possible. A fraction like \( \frac{20}{24} \) can be reduced to \( \frac{5}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 4 in this case.
Simplifying expressions allows for more accessible interpretation and paves the way for solving equations by making the structure of the mathematical relationships at play more evident.
To simplify an expression, always begin by combining like terms. For instance, in \( 3x + 5x \), you can combine the two terms to get \( 8x \). When dealing with fractions, reduce them to their smallest equivalent if possible. A fraction like \( \frac{20}{24} \) can be reduced to \( \frac{5}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 4 in this case.
Simplifying expressions allows for more accessible interpretation and paves the way for solving equations by making the structure of the mathematical relationships at play more evident.
Algebraic Fraction Subtraction
When it comes to algebraic fraction subtraction, it's key to handle the variables with the same care as the numbers. The process involves subtracting fractions with algebraic expressions in the numerator and/or the denominator—which can look intimidating at first glance. However, remember that the rules which apply to numerical fractions also apply here.
For the subtraction of fractions with algebraic denominators like in our initial exercise \( \frac{2}{3x - 1} - \frac{5x}{3x - 1} \), since the denominators are identical (\(3x - 1\)), subtract the numerators (\(2 - 5x\)), resulting in a new numerator over the shared denominator. The outcome is a single algebraic fraction representing the difference of the original fractions.
Algebraic fractions, like their numerical counterparts, can often be simplified further by factoring and reducing. This skill is particularly useful in calculus, where complex algebraic fractions frequently appear. Mastering this process makes the journey through higher mathematics significantly smoother.
For the subtraction of fractions with algebraic denominators like in our initial exercise \( \frac{2}{3x - 1} - \frac{5x}{3x - 1} \), since the denominators are identical (\(3x - 1\)), subtract the numerators (\(2 - 5x\)), resulting in a new numerator over the shared denominator. The outcome is a single algebraic fraction representing the difference of the original fractions.
Algebraic fractions, like their numerical counterparts, can often be simplified further by factoring and reducing. This skill is particularly useful in calculus, where complex algebraic fractions frequently appear. Mastering this process makes the journey through higher mathematics significantly smoother.
