The Importance of Eliminating Fractions

Oct 13, 2025 by miodeya.com 109 views

Understanding the Problem

When dealing with algebraic equations that contain fractions, it's often necessary to eliminate the fractions in order to simplify the equation and solve for the variable. In this case, we're given the equation $\frac{2y}{3} + \frac{1}{3} = \frac{y}{2} + \frac{1}{6}$ and we need to determine what value should be multiplied by the entire equation in order to eliminate the fractions.

The Importance of Eliminating Fractions

Eliminating fractions is a crucial step in solving algebraic equations, as it allows us to work with whole numbers and simplify the equation. When fractions are present, it can be difficult to isolate the variable and solve for its value. By eliminating the fractions, we can make the equation more manageable and increase our chances of finding the correct solution.

Finding the Least Common Multiple (LCM)

To eliminate the fractions in the given equation, we need to find the least common multiple (LCM) of the denominators. The denominators in this equation are 3, 3, 2, and 6. To find the LCM, we can list the multiples of each denominator and find the smallest number that appears in all the lists.

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...

The smallest number that appears in all the lists is 6, so the LCM of the denominators is 6.

Multiplying the Equation by the LCM

Now that we have found the LCM, we can multiply the entire equation by 6 to eliminate the fractions.

$\frac{2y}{3} + \frac{1}{3} = \frac{y}{2} + \frac{1}{6}$

$\times 6$

$6 \times \frac{2y}{3} + 6 \times \frac{1}{3} = 6 \times \frac{y}{2} + 6 \times \frac{1}{6}$

$4y + 2 = 3y + 1$

Simplifying the Equation

Now that we have eliminated the fractions, we can simplify the equation by combining like terms.

$4y + 2 = 3y + 1$

Subtracting 3y from both sides gives us:

$y + 2 = 1$

Subtracting 2 from both sides gives us:

$y = -1$

Conclusion

In this article, we have learned how to eliminate fractions in algebraic equations by finding the least common multiple (LCM) of the denominators and multiplying the entire equation by the LCM. We have applied this technique to the given equation $\frac{2y}{3} + \frac{1}{3} = \frac{y}{2} + \frac{1}{6}$ and simplified the equation to find the value of y. By following these steps, we can eliminate fractions and make algebraic equations more manageable.

Answer

Q: What is the purpose of eliminating fractions in algebraic equations?

A: The purpose of eliminating fractions in algebraic equations is to simplify the equation and make it easier to solve for the variable. When fractions are present, it can be difficult to isolate the variable and solve for its value. By eliminating the fractions, we can make the equation more manageable and increase our chances of finding the correct solution.

Q: How do I find the least common multiple (LCM) of the denominators?

A: To find the LCM of the denominators, you can list the multiples of each denominator and find the smallest number that appears in all the lists. Alternatively, you can use a formula or a calculator to find the LCM.

Q: What if the denominators have different prime factors?

A: If the denominators have different prime factors, you can use the following steps to find the LCM:

List the prime factors of each denominator.
Multiply the highest power of each prime factor together.
The result is the LCM of the denominators.

Q: Can I eliminate fractions by multiplying the equation by a different number?

A: No, you cannot eliminate fractions by multiplying the equation by a different number. The LCM of the denominators must be used to eliminate fractions. Multiplying the equation by a different number will not eliminate the fractions and may make the equation more complicated.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you can use the following steps to eliminate the fraction:

Multiply both sides of the equation by the variable in the denominator.
Simplify the equation to eliminate the fraction.

Q: Can I use a calculator to eliminate fractions?

A: Yes, you can use a calculator to eliminate fractions. Many calculators have a function to find the LCM of two or more numbers. You can also use a calculator to simplify fractions and eliminate them from the equation.

Q: What are some common mistakes to avoid when eliminating fractions?

A: Some common mistakes to avoid when eliminating fractions include:

Multiplying the equation by a different number instead of the LCM.
Not simplifying the equation after eliminating the fractions.
Not checking for errors in the LCM calculation.

Q: How do I check my work when eliminating fractions?

A: To check your work when eliminating fractions, you can:

Verify that the LCM of the denominators is correct.
Simplify the equation after eliminating the fractions.
Check for errors in the calculation.

Conclusion

In this article, we have answered some frequently asked questions about eliminating fractions in algebraic equations. We have covered topics such as finding the least common multiple (LCM) of the denominators, simplifying fractions, and checking work. By following these steps and avoiding common mistakes, you can eliminate fractions and make algebraic equations more manageable.