Problem 8 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{3}{x}+\frac{1}{5}=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solution set is \[ x = 10 \]

Step 1: Eliminate the fractions

Multiply every term by the least common denominator (LCD) of the fractions to eliminate the denominators. The denominators in this equation are 5, 2, and x. Thus, the LCD is 10x.

Step 2: Multiply both sides of the equation by the LCD

Multiply each term in the equation by 10x: \[ 10x \times \frac{3}{x} + 10x \times \frac{1}{5} = 10x \times \frac{1}{2} \]

Step 3: Simplify the equation

Simplify each term: \[ 10x \times \frac{3}{x} = 30 \] \[ 10x \times \frac{1}{5} = 2x \] \[ 10x \times \frac{1}{2} = 5x \] The equation becomes: \[ 30 + 2x = 5x \]

Step 4: Isolate the variable

Move all terms involving x to one side of the equation and constants to the other side: \[ 30 = 5x - 2x \]

Step 5: Solve for x

Combine like terms and solve for x: \[ 30 = 3x \] \[ x = \frac{30}{3} = 10 \]

Step 6: Verify the solution

Substitute x = 10 back into the original equation to verify: \[ \frac{3}{10} + \frac{1}{5} = \frac{1}{2} \] Simplify: \[ \frac{3}{10} + \frac{2}{10} = \frac{5}{10} = \frac{1}{2} \] The solution is verified.

Key Concepts

least common denominator

To solve a rational equation like \[\frac{3}{x} + \frac{1}{5} = \frac{1}{2}\], we first need to eliminate the fractions. The trick here is to find the Least Common Denominator (LCD), which is the smallest number that each denominator can divide into. In this equation, the denominators are 5, 2, and x. To find the LCD, look for the smallest value that each of these numbers can evenly divide into.

The steps are:

• Identify the denominators: 5, 2, and x.
• Find a common multiple: For 5 and 2, the smallest common multiple is 10. Since x can be any number, the LCD must include x.
• Combine multiples: The LCD is therefore 10x.

Multiplying every term in the equation by the LCD (10x) will eliminate the fractions. Each term in the equation gets multiplied by 10x:

• \(10x \times \frac{3}{x}\)
• \(10x \times \frac{1}{5}\)
• \(10x \times \frac{1}{2}\)

This removes all denominators and leaves us with a simpler equation in the next steps.

isolate the variable

After eliminating the fractions, we need to isolate the variable. Our equation now looks like this: 30 + 2x = 5x.

To isolate the variable (x), we must move all terms involving x to one side and constants to the other side. Here are the steps:

• Identify terms with the variable: In this case, 2x and 5x
• Subtract or add to isolate the variable: Move 2x to the right side by subtracting 2x from both sides: 30 = 5x - 2x

This simplifies our equation to:

• Combine like terms: 30 = 3x
• Divide both sides by 3 to solve for x: \(x = \frac{30}{3} = 10\)

Now, we have successfully isolated the variable and found that x = 10. The next step is to ensure this solution is correct by verifying it.

verify the solution

It's essential to verify the solution to ensure that x = 10 is indeed correct. We do this by plugging the value back into the original equation and checking if both sides equal.

1. Start with the original equation: \(\frac{3}{10} + \frac{1}{5} = \frac{1}{2}\)
2. Substitute x with 10:
\(\frac{3}{10} + \frac{1}{5}\)
3. Convert \(\frac{1}{5}\) to a common denominator:

\(\frac{1}{5} = \frac{2}{10}\)
4. Add the fractions:
\(\frac{3}{10} + \frac{2}{10} = \frac{5}{10}\)
5. Simplify the result:
\(\frac{5}{10} = \frac{1}{2}\)

Since both sides of the equation are equal (\(\frac{1}{2} = \frac{1}{2}\)), our solution x = 10 is verified. By verifying, we ensure our solution is accurate and we've followed the steps correctly.