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Chapter 4: Problem 49

Solve the equations by first clearing fractions. $$\frac{m}{3}+\frac{m}{6}=\frac{5}{9}$$

Short Answer

Expert verified
To isolate \(m\), multiply through by 18 and solve: \( m = \frac{10}{9} \).

Step by step solution

01

- Identify the Common Denominator

Determine the least common multiple (LCM) of the denominators 3, 6, and 9. The LCM of 3, 6, and 9 is 18.
02

- Clear the Fractions

To eliminate the fractions, multiply both sides of the equation by the common denominator (18): \[ 18 \times \frac{m}{3} + 18 \times \frac{m}{6} = 18 \times \frac{5}{9} \]
03

- Simplify Each Term

Distribute the 18 across each term: \[ 18 \times \frac{m}{3} = 6m \] \[ 18 \times \frac{m}{6} = 3m \] \[ 18 \times \frac{5}{9} = 10 \]
04

- Combine Like Terms

Combine the terms involving \(m\): \[ 6m + 3m = 9m \] The equation becomes: \[ 9m = 10 \]
05

- Solve for \(m\)

Isolate \(m\) by dividing both sides by 9: \[ m = \frac{10}{9} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominators
A common denominator is a shared multiple of the denominators of several fractions. It's important because it allows you to combine fractions easily. For instance, in the equation \(\frac{m}{3} + \frac{m}{6} = \frac{5}{9}\) , the denominators are 3, 6, and 9. To combine these fractions, we need a common denominator.

To find a common denominator, first determine the least common multiple (LCM) of the denominators.

In our example:
  • The LCM of 3, 6, and 9 is 18
By using 18 as the common denominator, you can easily clear the fractions.
Clearing Fractions
Clearing fractions means eliminating the fractions by converting them into whole numbers for simplicity. This can be achieved by multiplying each term by the common denominator.

In our equation: \(\frac{m}{3} + \frac{m}{6} = \frac{5}{9}\)

We multiply every term by the common denominator (18):
\[ 18 \times \frac{m}{3} + 18 \times \frac{m}{6} = 18 \times \frac{5}{9} \]

This removes the fractions and simplifies calculation.
Least Common Multiple
The Least Common Multiple (LCM) is the smallest number that is a multiple of each denominator. Finding the LCM is crucial for operations involving multiple fractions.

For the denominators 3, 6, and 9, we find:
  • Multiples of 3: 3, 6, 9, 12, 15, 18...
  • Multiples of 6: 6, 12, 18...
  • Multiples of 9: 9, 18...
The smallest common multiple of these numbers is 18. This is why 18 is chosen to multiply through the fractions, simplifying the equation.
Combining Like Terms
Combining like terms means adding or subtracting terms that have the same variable and power.

After clearing the fractions in our example, we have:
\[ 6m + 3m = 10 \]

The terms \(6m\) and \(3m\) are like terms as they both contain the variable \(m\). Adding them gives:

\[ 9m \]

This simplifies the equation to:
\[ 9m = 10 \]
Isolating Variables
Isolating the variable means manipulating the equation to get the variable by itself on one side of the equation, which makes it easier to solve.

In our simplified equation:
\[ 9m = 10 \]

We isolate \(m\) by dividing both sides by 9:
  • \( m = \frac{10}{9} \)
The solution \( m = \frac{10}{9} \) gives the value of \( m \) that satisfies the original equation.

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