Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-20,\) solve each equation and check. $$ \frac{x}{5}-\frac{x}{10}=7 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 70\).

Step 1: Equalize the Denominators

The first task is to equalize the denominators in the equation. The fractions involved are \(\frac{x}{5}\) and \(\frac{x}{10}\). The least common denominator (LCD) between 5 and 10 is 10. Therefore, rewrite \(\frac{x}{5}\) so it has a denominator of 10: \(\frac{x}{5} = \frac{2x}{10}\). So the equation becomes \(\frac{2x}{10} - \frac{x}{10} = 7\).

Step 2: Simplify the Equation

Now that both fractions have the same denominator, we can combine them: \(\frac{2x}{10} - \frac{x}{10} = \frac{2x - x}{10} = \frac{x}{10}\). The equation thus becomes \(\frac{x}{10} = 7\).

Step 3: Solve for x

To solve for \(x\), multiply both sides of the equation by 10 to isolate \(x\): \(x = 7 \times 10\). Therefore, \(x = 70\).

Step 4: Check the Solution

Substitute \(x = 70\) back into the original equation to verify the solution: \(\frac{70}{5} - \frac{70}{10} = 14 - 7 = 7\). This simplifies to 7, which matches the right side of the equation, confirming that the solution is correct.

Key Concepts

Least Common Denominator

When solving equations with fractions, finding the Least Common Denominator (LCD) is a critical first step. The LCD is the smallest number that can be exactly divided by all the denominators in the equation. In this exercise, we have the denominators 5 and 10.
- To find the LCD: - List multiples of each denominator. For 5: 5, 10, 15,... For 10: 10, 20, 30,... - The smallest multiple they share is 10.By rewriting the fractions to have the same denominator, we make them easier to work with. In the equation \(\frac{x}{5} - \frac{x}{10} = 7\), rewrite \(\frac{x}{5}\) as \(\frac{2x}{10}\) because multiplying the numerator and denominator by 2 results in a denominator of 10. Now both terms can be combined as one.

Combining Fractions

Once the denominators are the same, combining fractions is straightforward. We use the rule that states: if two fractions have the same denominator, you can directly add or subtract the numerators.
- Consider the modified equation \(\frac{2x}{10} - \frac{x}{10} = 7\). - The denominators are both 10, so we subtract the numerators: \(2x - x = x\).This operation leaves a simple fraction \(\frac{x}{10} = 7\), setting up a much easier equation to solve. Simplifying fractions by combining them helps organize terms and simplifies the pathway to isolating the variable.

Checking Solutions

Checking your solution is an essential part of solving equations, particularly when fractions are involved. Substituting the solution back into the original equation ensures accuracy.
- Our solution for \(x\) is 70. - Substitute it into the original equation: \(\frac{70}{5} - \frac{70}{10} = 7\). - Calculate: \(14 - 7 = 7\).The left side of the equation equals the right side, showing that \(x = 70\) satisfies the equation. This check confirms that our steps are correct and the answer is valid. Checking is a good habit that can prevent errors in math problems.