What is the least common denominator that can be used to solve this equation?
[tex]$\frac{1}{5}+\frac{2}{x-3}=5$[/tex]
A. [tex]$x(x-3)$[/tex]
B. [tex]$5 x(x-3)$[/tex]
C. [tex]$x$[/tex]
D. [tex]$x-3$[/tex]
Answer (2)
Identify the denominators in the equation: 5 and x − 3 .
Determine the least common multiple (LCM) of the denominators, which is 5 ( x − 3 ) .
Compare the result with the given options and choose the correct one.
The least common denominator among the given options is 5 x ( x − 3 ) .
Explanation
Understanding the Problem The problem asks for the least common denominator (LCD) of the equation 5 1 + x − 3 2 = 5 . The LCD is the least common multiple (LCM) of the denominators in the equation.
Finding the LCD The denominators in the equation are 5 and x − 3 . Since 5 is a constant and x − 3 is an algebraic expression, their least common multiple is simply their product.
Determining the LCD The least common denominator is therefore 5 ( x − 3 ) .
Analyzing the Options Comparing 5 ( x − 3 ) with the given options, we see that option B, 5 x ( x − 3 ) , is not the least common denominator because it includes an extra factor of x . Option A, x ( x − 3 ) , is also not the least common denominator because it is missing the factor of 5. Option C, x , and option D, x − 3 , are not the LCD either. However, if we multiply the entire equation by 5 ( x − 3 ) , we can eliminate the fractions. So, we need to find the option that represents the least common denominator.
Finding the Correct Option The correct least common denominator is 5 ( x − 3 ) . However, this is not one of the options. Let's re-examine the problem and the options. We have the equation 5 1 + x − 3 2 = 5 . The denominators are 5 and x − 3 . The least common denominator must be a multiple of both 5 and x − 3 . Among the given options, only 5 x ( x − 3 ) is a multiple of both 5 and x − 3 . However, we are looking for the least common denominator. Multiplying the equation by 5 ( x − 3 ) would eliminate the fractions. Multiplying by 5 x ( x − 3 ) would also eliminate the fractions, but it's not the least common denominator. The question is a bit ambiguous, but we are looking for the expression that, when multiplied by each term, will eliminate the denominators. In this case, 5 ( x − 3 ) is the LCD. However, since it is not an option, let's consider the given options. Option A: x ( x − 3 ) . If we multiply the equation by x ( x − 3 ) , we get 5 x ( x − 3 ) + x − 3 2 x ( x − 3 ) = 5 x ( x − 3 ) , which simplifies to 5 x ( x − 3 ) + 2 x = 5 x ( x − 3 ) . The first term still has a denominator of 5, so this is not the LCD. Option B: 5 x ( x − 3 ) . If we multiply the equation by 5 x ( x − 3 ) , we get 5 5 x ( x − 3 ) + x − 3 2 ( 5 x ) ( x − 3 ) = 5 ( 5 x ) ( x − 3 ) , which simplifies to x ( x − 3 ) + 10 x = 25 x ( x − 3 ) . This eliminates the denominators. Option C: x . If we multiply the equation by x , we get 5 x + x − 3 2 x = 5 x . Neither denominator is eliminated. Option D: x − 3 . If we multiply the equation by x − 3 , we get 5 x − 3 + 2 = 5 ( x − 3 ) . The first term still has a denominator of 5. Therefore, the least common denominator among the given options is 5 x ( x − 3 ) .
Final Answer The least common denominator that can be used to solve the equation is 5 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 least common denominator helps you find a common unit to accurately add the ingredients, ensuring your cake turns out perfectly. Similarly, in engineering, when calculating stress on a structure with forces distributed as fractions, using the LCD simplifies the calculations and ensures accurate results.
The least common denominator for the equation 5 1 + x − 3 2 = 5 is found to be 5 x ( x − 3 ) , which covers all fractional components of the equation. Among the provided options, option B is the correct choice as it allows for the elimination of fractions when multiplied. Thus, the correct answer is B.
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