Simplify: [tex]$\frac{54 k-6}{3 k+1}$[/tex]
Answer (2)
Factor out a constant from the numerator: 54 k − 6 = 6 ( 9 k − 1 ) .
Rewrite the expression: 3 k + 1 6 ( 9 k − 1 ) .
Express 9 k − 1 in terms of 3 k + 1 : 9 k − 1 = 3 ( 3 k + 1 ) − 4 .
Simplify the expression: 3 k + 1 6 ( 3 ( 3 k + 1 ) − 4 ) = 18 − 3 k + 1 24 .
18 − 3 k + 1 24
Explanation
Problem Analysis We are given the expression 3 k + 1 54 k − 6 and our goal is to simplify it.
Factoring the Numerator First, we can factor out a 6 from the numerator: 54 k − 6 = 6 ( 9 k − 1 ) So the expression becomes: 3 k + 1 6 ( 9 k − 1 )
Rewriting the Numerator Now, we want to express the numerator in terms of ( 3 k + 1 ) so we can simplify the fraction. Notice that 9 k − 1 can be rewritten as: 9 k − 1 = 3 ( 3 k ) − 1 = 3 ( 3 k + 1 ) − 3 − 1 = 3 ( 3 k + 1 ) − 4 So, we can rewrite the expression as: 3 k + 1 6 ( 3 ( 3 k + 1 ) − 4 )
Distributing the Constant Now, we distribute the 6 in the numerator: 3 k + 1 6 ( 3 ( 3 k + 1 ) − 4 ) = 3 k + 1 18 ( 3 k + 1 ) − 24
Simplifying the Expression We can now split the fraction into two parts: 3 k + 1 18 ( 3 k + 1 ) − 24 = 3 k + 1 18 ( 3 k + 1 ) − 3 k + 1 24 Since 3 k + 1 18 ( 3 k + 1 ) = 18 , the expression simplifies to: 18 − 3 k + 1 24
Final Answer Therefore, the simplified expression is: 18 − 3 k + 1 24
Examples
Consider a scenario where you are distributing flyers in a neighborhood. You have 54 k − 6 flyers to distribute among 3 k + 1 houses. Simplifying the expression 3 k + 1 54 k − 6 helps you understand how many flyers each house gets on average. The simplified form 18 − 3 k + 1 24 tells you that each house gets approximately 18 flyers, but there's a correction factor depending on the value of k . This kind of simplification is useful in resource allocation and understanding average distribution rates.
To simplify the expression 3 k + 1 54 k − 6 , we factor the numerator to get 3 k + 1 6 ( 9 k − 1 ) , rewrite the numerator, and then split the fraction to obtain the final simplified form of 18 − 3 k + 1 24 .
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