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In Mathematics / College | 2025-07-05

What is the least common denominator that can be used to solve this equation?

[tex]\frac{1}{x}+\frac{2}{x-3}=5[/tex]

A. [tex]$5 x(x-3)$[/tex]
B. [tex]$x$[/tex]
C. [tex]$x(x-3)$[/tex]
D. [tex]$x-3$[/tex]

Asked by lilmama8294tooka

Answer (2)

Identify the denominators in the equation: x and x − 3 .
Find the least common multiple (LCM) of the denominators, which is x ( x − 3 ) .
The least common denominator (LCD) is the same as the LCM, so it is x ( x − 3 ) .
The correct answer is x ( x − 3 ) ​ .

Explanation

Understanding the Problem We are given the equation x 1 ​ + x − 3 2 ​ = 5 and asked to find the least common denominator (LCD) that can be used to solve it. The LCD is the least common multiple (LCM) of the denominators in the equation.

Finding the Least Common Multiple The denominators in the equation are x and x − 3 . To find the LCM, we look for the smallest expression that is divisible by both x and x − 3 . Since x and x − 3 do not share any common factors, their LCM is simply their product.

Determining the LCD The least common multiple of x and x − 3 is x ( x − 3 ) . Therefore, the least common denominator that can be used to solve the equation is x ( x − 3 ) .

Selecting the Correct Option Comparing our result with the given options: A. 5 x ( x − 3 ) B. x C. x ( x − 3 ) D. x − 3 We see that option C, x ( x − 3 ) , matches our calculated LCD.

Final Answer Therefore, the least common denominator that can be used to solve the equation is x ( x − 3 ) ​ .


Examples
When solving equations involving fractions, finding the least common denominator is crucial. Imagine you're baking a cake and need to combine different ingredients measured in fractions. The LCD helps you find a common unit to accurately measure and combine these ingredients, ensuring the cake turns out perfectly. Similarly, in various fields like engineering, finance, and computer science, dealing with fractional quantities requires finding the LCD to perform calculations accurately and efficiently. For example, when calculating the total resistance of resistors connected in parallel, you need to find the LCD of the reciprocals of the resistances.

Answered by GinnyAnswer | 2025-07-05

The least common denominator (LCD) for the equation x 1 ​ + x − 3 2 ​ = 5 is x ( x − 3 ) . Therefore, the correct answer is option C: x ( x − 3 ) .
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Answered by Anonymous | 2025-07-19

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