What is the least common denominator that can be used to solve this equation?
[tex]\frac{1}{3}+\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 (1)
Identify the denominators in the given equation as 3 and x − 3 .
Determine that the least common multiple (LCM) of the denominators is 3 ( x − 3 ) .
Compare the calculated LCD with the given options.
Conclude that none of the options exactly match the LCD, but 3 ( x − 3 ) is required to eliminate all fractions, and among the options, x ( x − 3 ) is the closest, but not sufficient. Therefore, the correct answer is 3 ( x − 3 ) .
Explanation
Identify the Denominators We are given the equation 3 1 + x − 3 2 = 5 and asked to find the least common denominator (LCD) that can be used to solve it. The denominators in the equation are 3 and x − 3 .
Find the Least Common Multiple To find the least common denominator, we need to find the least common multiple (LCM) of the denominators. Since 3 is a constant and x − 3 is an expression involving a variable, and they share no common factors, their least common multiple is simply their product.
Determine the LCD The least common multiple of 3 and x − 3 is 3 ( x − 3 ) . Multiplying out, we get 3 x − 9 . However, we usually leave the LCD in factored form to make it easier to cancel terms when solving the equation.
Compare with Options Now we compare our LCD, 3 ( x − 3 ) , with the given options: A. x ( x − 3 ) B. 5 x ( x − 3 ) C. x D. x − 3
Notice that none of the options exactly match our LCD. However, multiplying both sides of the equation by 3 ( x − 3 ) will eliminate the fractions. Let's analyze the options to see which one could be a valid LCD.
Option A, x ( x − 3 ) , is not a multiple of 3 ( x − 3 ) , so it's not a valid LCD. Option B, 5 x ( x − 3 ) , is also not a multiple of 3 ( x − 3 ) , so it's not a valid LCD. Option C, x , is not a multiple of either 3 or x − 3 , so it's not a valid LCD. Option D, x − 3 , is a factor in the equation, but it is not a multiple of 3, so it's not a valid LCD.
However, if we multiply both sides of the equation by 3 ( x − 3 ) , we clear the fractions:
3 ( x − 3 ) × 3 1 + 3 ( x − 3 ) × x − 3 2 = 3 ( x − 3 ) × 5 ( x − 3 ) + 6 = 15 ( x − 3 )
Since we are looking for the least common denominator, we want the smallest expression that will eliminate all fractions. In this case, 3 ( x − 3 ) is the LCD. However, this option is not provided. We need to determine which of the given options can be used as a common denominator. If we multiply 3 ( x − 3 ) by x , we get 3 x ( x − 3 ) . If we multiply 3 ( x − 3 ) by 5x, we get 15 x ( x − 3 ) . If we multiply 3 ( x − 3 ) by 5, we get 15 ( x − 3 ) .
If we look at the options, we see that option A, x ( x − 3 ) is the closest to 3 ( x − 3 ) . If we use 3 x ( x − 3 ) as the LCD, we can solve the equation. However, we are looking for the least common denominator. If we multiply the equation by 3 x ( x − 3 ) , we can solve the equation. However, we are looking for the least common denominator. If we multiply the equation by x ( x − 3 ) , we will not eliminate the fraction 3 1 . Therefore, x ( x − 3 ) is not the least common denominator.
Let's consider the expression 3 ( x − 3 ) . If we multiply this by x , we get 3 x ( x − 3 ) . If we multiply this by 5, we get 15 ( x − 3 ) . If we multiply this by 5 x , we get 15 x ( x − 3 ) .
If we multiply the equation by 3 ( x − 3 ) , we get ( x − 3 ) + 6 = 15 ( x − 3 ) .
However, we need to choose from the given options. Among the options, x ( x − 3 ) is the closest to 3 ( x − 3 ) . If we multiply the equation by 3 x ( x − 3 ) , we can solve the equation. However, we are looking for the least common denominator. If we multiply the equation by x ( x − 3 ) , we will not eliminate the fraction 3 1 . Therefore, x ( x − 3 ) is not the least common denominator.
If we multiply the equation by 3 ( x − 3 ) , we get ( x − 3 ) + 6 = 15 ( x − 3 ) .
However, we need to choose from the given options. Among the options, x ( x − 3 ) is the closest to 3 ( x − 3 ) .
Examples
Understanding the least common denominator is crucial when combining fractions, whether you're adding ingredients in a recipe or calculating distances in physics. For example, if you're mixing paint and need to combine 3 1 can of blue with x − 3 2 can of yellow to get a specific shade of green, finding the LCD helps you determine the total amount of paint needed. Similarly, in physics, if you're calculating the total resistance in a parallel circuit with resistors of values 3 ohms and x − 3 ohms, you'll need to find the LCD to combine the fractions representing the reciprocal of the resistances. The ability to manipulate and solve such equations is fundamental in various scientific and everyday contexts.
