Problem 14 - Elementary Algebra

Question

For exercises 11-30, (a) solve. (b) check. $$ \frac{13}{d}-\frac{5}{9}=\frac{1}{6} $$

Step-by-Step Solution

Answer

d = 18

Step 1 - Isolate the variable term

Start by isolating the term with the variable. Add \ \( \frac{5}{9} \) to both sides to get: \ \( \frac{13}{d} = \frac{1}{6} + \frac{5}{9} \).

Step 2 - Find a common denominator

The common denominator for \ \(6\) and \ \(9\) is \ \(18\). Convert the fractions to have this common denominator: \ \( \frac{1}{6} = \frac{3}{18} \) and \ \( \frac{5}{9} = \frac{10}{18} \).

Step 3 - Add the fractions

Add the two fractions: \ \( \frac{3}{18} + \frac{10}{18} = \frac{13}{18} \). So, \ \( \frac{13}{d} = \frac{13}{18} \).

Step 4 - Solve for d

To find \ \( d \), cross-multiply to get: \ \( 13 \times 18 = 13 \times d \). Thus, \ \( d = 18 \).

Step 5 - Check the solution

Substitute \ \( d = 18 \) back into the original equation to verify: \ \( \frac{13}{18} - \frac{5}{9} = \frac{1}{6} \). Simplify the left side: \ \( \frac{13}{18} - \frac{10}{18} = \frac{3}{18} = \frac{1}{6} \). The equation holds true.

Key Concepts

isolate variable

When solving rational equations, the first step often involves isolating the variable. In the given exercise, we have the equation: \[ \frac{13}{d} - \frac{5}{9} = \frac{1}{6} \] Our goal is to get the term with the variable, in this case, \( \frac{13}{d} \), by itself on one side of the equation. We do this by adding \( \frac{5}{9} \) to both sides: \[ \frac{13}{d} = \frac{1}{6} + \frac{5}{9} \] This isolates the variable term \( \frac{13}{d} \) and simplifies our equation, making it easier to solve in the next steps. Always remember, isolating the variable helps you focus on one unknown at a time.

common denominator

To add or subtract fractions, they need a common denominator. In our equation \( \frac{13}{d} = \frac{1}{6} + \frac{5}{9} \), we must find the least common multiple (LCM) of 6 and 9. The LCM is 18. We then convert each fraction to have 18 as the denominator: \[ \frac{1}{6} = \frac{3}{18} \] \[ \frac{5}{9} = \frac{10}{18} \] Now, add the fractions on the right side: \[ \frac{3}{18} + \frac{10}{18} = \frac{13}{18} \] This simplifies our equation to: \[ \frac{13}{d} = \frac{13}{18} \] Common denominators are crucial because they allow us to combine fractions seamlessly, leading to simpler equations.

cross-multiplication

Cross-multiplication is a common method to solve equations involving fractions. From our previous step, we have: \[ \frac{13}{d} = \frac{13}{18} \] To find \( d \), cross-multiply the fractions: \[ 13 \times 18 = 13 \times d \] This simplifies to: \[ 234 = 13d \] By dividing both sides by 13, we isolate \( d \): \[ d = \frac{234}{13} = 18 \] Cross-multiplication allows us to clear the fractions, transforming our equation into a simpler form that is easier to solve.

checking solutions

Always verify your solution by substituting it back into the original equation. In this case, we found \( d = 18 \). Let's substitute \( d \) back into the original equation: \[ \frac{13}{18} - \frac{5}{9} = \frac{1}{6} \] First, convert \( \frac{5}{9} \) to \( \frac{10}{18} \): \[ \frac{13}{18} - \frac{10}{18} = \frac{3}{18} = \frac{1}{6} \] The left side simplifies to the right side, confirming our solution is correct. Checking solutions ensures our answer is accurate and the original equation holds true.