What can each term of the equation be multiplied by to eliminate fractions?
[tex]$\frac{1}{2} x-\frac{5}{4}+2 x=\frac{5}{6}+x$[/tex]
Answer (2)
To eliminate fractions from the equation 2 1 x − 4 5 + 2 x = 6 5 + x , we need to find the least common multiple (LCM) of the denominators 2, 4, and 6.
Calculate the LCM of 2, 4, and 6, which is 12.
Multiply each term of the equation by 12 to eliminate the fractions.
The equation becomes 6 x − 15 + 24 x = 10 + 12 x .
The number to multiply each term by to eliminate fractions is 12 .
Explanation
Understanding the Problem We are given the equation 2 1 x − 4 5 + 2 x = 6 5 + x Our goal is to find a number that we can multiply each term of the equation by to eliminate the fractions. This number should be a common multiple of the denominators of the fractions in the equation.
Finding the Least Common Multiple (LCM) The denominators in the equation are 2, 4, and 6. To eliminate the fractions, we need to find the least common multiple (LCM) of these denominators. The LCM is the smallest number that is a multiple of all the denominators.
Calculating the LCM To find the LCM of 2, 4, and 6, we can list the multiples of each number: Multiples of 2: 2, 4, 6, 8, 10, 12, 14, ... Multiples of 4: 4, 8, 12, 16, 20, ... Multiples of 6: 6, 12, 18, 24, ... The smallest number that appears in all three lists is 12. Therefore, the LCM of 2, 4, and 6 is 12. Alternatively, we can use prime factorization: 2 = 2 4 = 2^2 6 = 2 * 3 LCM(2, 4, 6) = 2^2 * 3 = 4 * 3 = 12
Multiplying by the LCM Multiplying each term of the equation by 12 will eliminate the fractions: 12 ⋅ 2 1 x − 12 ⋅ 4 5 + 12 ⋅ 2 x = 12 ⋅ 6 5 + 12 ⋅ x 6 x − 15 + 24 x = 10 + 12 x
Conclusion Therefore, each term of the equation can be multiplied by 12 to eliminate the fractions. The correct answer is 12.
Examples
When you're cooking and need to adjust a recipe that uses fractional measurements, finding the least common multiple helps you scale the ingredients correctly. For example, if a recipe calls for 2 1 cup of flour, 4 1 teaspoon of salt, and 3 1 cup of sugar, you can multiply all the ingredients by the LCM of 2, 4, and 3 (which is 12) to easily scale the recipe up or down while maintaining the correct proportions. This ensures your dish tastes as intended, whether you're making a small batch or enough to feed a crowd.
To eliminate fractions from the equation 2 1 x − 4 5 + 2 x = 6 5 + x , multiply each term by 12, the least common multiple of the denominators 2, 4, and 6. This results in a new equation without fractions, making it easier to solve. Therefore, the number to multiply by is 12 .
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