Problem 19 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{x}{18}=\frac{1}{3}-\frac{x}{2} $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{5}\).

Step 1: Eliminate Fractions

To eliminate fractions, find a common denominator for the terms on both sides of the equation. Here, the common denominator for 18, 3, and 2 is 18. Multiply every term by 18 to get rid of the fractions.\[ 18 \cdot \frac{x}{18} = 18 \left( \frac{1}{3} - \frac{x}{2} \right) \] Simplifying gives: \[ x = 6 - 9x \]

Step 2: Rearrange the Equation

Add \(9x\) to both sides to bring all 'x' terms to one side of the equation: \[ x + 9x = 6 \] This simplifies to: \[ 10x = 6 \]

Step 3: Solve for x

Divide both sides by 10 to solve for \(x\): \[ x = \frac{6}{10} = \frac{3}{5} \]

Step 4: Check the Result

Substitute \(x = \frac{3}{5}\) back into the original equation to verify the solution: \[ \frac{3/5}{18} = \frac{1}{3} - \frac{3/5}{2} \] Simplify each term: \[ \frac{1}{30} = \frac{1}{3} - \frac{3}{10} \] Convert to common denominators and check: \[ \frac{1}{30} = \frac{10}{30} - \frac{9}{30} \] Solving, \[ \frac{1}{30} = \frac{1}{30} \] Thus, the solution is verified.

Key Concepts

Eliminating Fractions

Eliminating fractions is an important step when solving equations involving fractions. Fractions can make equations look more complex than they are, but don't worry! To eliminate fractions, you look for a number that can work as a common multiple for all the denominators in the equation. This number is called the "common denominator." For example, in the equation \( \frac{x}{18} = \frac{1}{3} - \frac{x}{2} \), the denominators are 18, 3, and 2. The smallest common denominator among these numbers is 18.Once you have the common denominator, multiply each term in the equation by this number. This step will transform all the fractions into whole numbers, making it much more straightforward to solve the equation:

• Multiply \(\frac{x}{18}\) by 18 to get \(x\).
• Multiply \(\frac{1}{3}\) by 18 to get 6.
• Multiply \(\frac{x}{2}\) by 18 to get 9x.

The equation becomes \( x = 6 - 9x \). Now, it's just a regular equation, and you can proceed with solving it further.

Common Denominator

A common denominator is a number that serves as a shared multiple of the denominators in a set of fractions. Finding a common denominator is crucial when you perform operations like addition, subtraction, and equation solving with fractions.In our example equation \( \frac{x}{18} = \frac{1}{3} - \frac{x}{2} \), the denominators are 18, 3, and 2. To handle these fractions, we want the denominators to "fit" into one number, and that number should be as small as possible, so it's easier to work with. For these numbers, the smallest common denominator is 18:

• 18 is already a denominator in the equation.
• 3 can go into 18 because \(18 \div 3 = 6\).
• 2 fits into 18 because \(18 \div 2 = 9\).

Multiplying the entire equation by the common denominator (18) clears all fractions. This makes your equation simpler and more straightforward to solve. It's similar to making a road smoother before driving on it—it eases the entire process!

Verifying Solutions

After finding a solution to an equation, it's essential to verify that it truly works. Verification is like double-checking your work to ensure you've got the correct answer. This is crucial, especially when dealing with complex equations or when fractions are involved.To verify the solution, substitute the value back into the original equation. Let's check our solution of \( x = \frac{3}{5} \) from the equation \( \frac{x}{18} = \frac{1}{3} - \frac{x}{2} \):

• Substitute \( x = \frac{3}{5} \) into \( \frac{x}{18} \) and \( \frac{x}{2} \).
• The left side becomes \( \frac{3/5}{18} = \frac{1}{30} \).
• The right side simplifies by converting all fractions to have the same denominator, \( \frac{10}{30} - \frac{9}{30} = \frac{1}{30} \).

Since both sides are equal (\( \frac{1}{30} = \frac{1}{30} \)), the solution \( x = \frac{3}{5} \) is verified. Ensuring your solution satisfies the original equation will increase your confidence and accuracy in solving mathematical problems.