Eliminate Fractions: Find The Right Multiplier!

by Andrew McMorgan 48 views

Hey guys! Let's break down this math problem together. We've got an equation with fractions, and nobody wants to deal with those when solving for x. So, the big question is: what's the magic number we can multiply every single term by to make those fractions disappear?

Understanding the Problem

Our equation looks like this:

12xβˆ’54Γ·2x=56+x\frac{1}{2} x-\frac{5}{4} \div 2 x=\frac{5}{6}+x

Before we jump into finding the multiplier, let's simplify that division part. Remember that dividing by a number is the same as multiplying by its reciprocal. So, 54Γ·2x\frac{5}{4} \div 2x is the same as 54βˆ—12x\frac{5}{4} * \frac{1}{2x} which simplifies to 58x\frac{5}{8x}.

Now, our equation is:

12xβˆ’58x=56+x\frac{1}{2} x - \frac{5}{8x} = \frac{5}{6} + x

To get rid of these fractions, we need to find the least common multiple (LCM) of the denominators. The denominators we have are 2, 8x, and 6. Finding the LCM will give us the smallest number that all the denominators can divide into evenly. This is important because multiplying each term by the LCM will clear out all the fractions, making the equation much easier to solve.

So, what's the LCM of 2, 8x, and 6? First, let's find the LCM of the numbers 2, 8, and 6. The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24... The multiples of 8 are 8, 16, 24, 32... The multiples of 6 are 6, 12, 18, 24, 30.... The least common multiple of 2, 8, and 6 is 24. Since we also have an 'x' in the denominator of one of our terms, we must also include 'x' in our least common multiple. So, the LCM is 24x.

Therefore, to eliminate the fractions, we have to consider 24x which is not an option in the answers, and the equation has not been correctly written. Let's proceed with the original equation which is:

12xβˆ’54βˆ—2x=56+x\frac{1}{2} x-\frac{5}{4} * 2 x=\frac{5}{6}+x

12xβˆ’104x=56+x\frac{1}{2} x-\frac{10}{4} x=\frac{5}{6}+x

12xβˆ’52x=56+x\frac{1}{2} x-\frac{5}{2} x=\frac{5}{6}+x

The denominators we have are 2 and 6. To get rid of these fractions, we need to find the least common multiple (LCM) of the denominators. Finding the LCM will give us the smallest number that all the denominators can divide into evenly. This is important because multiplying each term by the LCM will clear out all the fractions, making the equation much easier to solve.

The multiples of 2 are: 2, 4, 6, 8, 10, 12... The multiples of 6 are: 6, 12, 18, 24...

The least common multiple of 2 and 6 is 6. So, our magic number is 6!

Why This Works

Think of it like this: each fraction has a denominator that's dividing the numerator. To undo that division, we need to multiply by something that's a multiple of that denominator. The LCM is the smallest such number that works for all the denominators in the equation. This keeps the numbers manageable and avoids unnecessary complications.

Applying the Multiplier

Let's see what happens when we multiply each term in the original equation by 6:

6βˆ—(12x)βˆ’6βˆ—(52x)=6βˆ—(56)+6βˆ—(x)6 * (\frac{1}{2}x) - 6 * (\frac{5}{2}x) = 6 * (\frac{5}{6}) + 6 * (x)

This simplifies to:

3xβˆ’15x=5+6x3x - 15x = 5 + 6x

See? No more fractions! The equation is now much easier to solve using standard algebraic techniques.

The Answer

So, the answer to the question "What can each term of the equation be multiplied by to eliminate the fractions before solving?" is:

B. 6

Why Not the Other Options?

Let's quickly look at why the other options aren't the best choice:

  • A. 2: While multiplying by 2 would get rid of the fraction with a denominator of 2, it wouldn't eliminate the fraction with a denominator of 6.
  • C. 10: Multiplying by 10 wouldn't eliminate the fractions with denominators of 2 or 6.
  • D. 12: While multiplying by 12 would eliminate all the fractions (since 12 is a multiple of both 2 and 6), it's not the least common multiple. Using 6 keeps the numbers smaller and easier to work with.

Key Takeaways for Fraction Elimination

Alright, let's nail down the key steps for making those pesky fractions vanish from your equations:

  1. Identify all the denominators: Look at each fraction in your equation and write down the numbers on the bottom (the denominators). In our example, these were 2 and 6.
  2. Find the Least Common Multiple (LCM): Determine the smallest number that each of your denominators can divide into evenly. This is your magic multiplier! If you are not sure how to do this, list the multiples of each denominator until you find a common multiple. The smallest one is the LCM.
  3. Multiply Every Term: This is crucial. Multiply every single term in the entire equation (on both sides of the equals sign) by the LCM you just found. Don't miss any!
  4. Simplify: After multiplying, simplify each term. The denominators should cancel out, leaving you with an equation free of fractions.

Example Time!

Let's say you have the equation:

x3+14=56\frac{x}{3} + \frac{1}{4} = \frac{5}{6}

  1. Denominators: 3, 4, and 6
  2. LCM: The LCM of 3, 4, and 6 is 12.
  3. Multiply: Multiply every term by 12: 12βˆ—(x3)+12βˆ—(14)=12βˆ—(56)12 * (\frac{x}{3}) + 12 * (\frac{1}{4}) = 12 * (\frac{5}{6})
  4. Simplify: 4x+3=104x + 3 = 10

Boom! Fractions are gone, and you're left with a much simpler equation.

Common Mistakes to Avoid

  • Forgetting to Multiply Every Term: This is the biggest mistake people make. If you only multiply some terms, you're changing the equation and won't get the right answer. Treat each term like it's begging for that LCM!
  • Using a Common Multiple Instead of the LCM: While any common multiple will eliminate the fractions, using the least common multiple keeps the numbers smaller and easier to manage. Why make your life harder than it needs to be?
  • Incorrectly Calculating the LCM: Double-check your LCM calculations! A mistake here will throw off the entire process.
  • Combining Terms Too Early: Don't start combining terms until after you've eliminated the fractions. Focus on getting rid of those fractions first, then simplify.

Practice Makes Perfect

The best way to master this technique is to practice! Find some equations with fractions and work through them. The more you practice, the faster and more confident you'll become.

So there you have it! Clearing fractions doesn't have to be scary. Find that LCM, multiply away, and enjoy your fraction-free equation-solving experience. You got this!