Problem 44 - Elementary Algebra

Question

For exercises 43-58, (a) solve. (b) check. $$ \frac{3}{k}+\frac{7}{18}=\frac{5}{9} $$

Step-by-Step Solution

Answer

k = 54

Step 1: Isolate the Variable

First, we need to get all terms involving the variable k on one side of the equation. Subtract \(\frac{7}{18}\) from both sides:\[ \frac{3}{k} = \frac{5}{9} - \frac{7}{18} \]

Step 2: Find a Common Denominator

To subtract the fractions on the right-hand side, find a common denominator. The common denominator of 9 and 18 is 18:\[ \frac{5}{9} = \frac{10}{18} \]Thus,\[ \frac{3}{k} = \frac{10}{18} - \frac{7}{18} \]

Step 3: Simplify the Right-Hand Side

Subtract the fractions on the right-hand side:\[ \frac{3}{k} = \frac{10 - 7}{18} = \frac{3}{18} \]

Step 4: Cross-Multiply

Now, cross-multiply to solve for k:\[ 3 \times 18 = k \times 3 \]This simplifies to:\[ 54 = k \]

Step 5: Check the Solution

Substitute k = 54 back into the original equation to verify:\[ \frac{3}{54} + \frac{7}{18} = \frac{5}{9} \]Simplify the fractions on the left-hand side:\[ \frac{1}{18} + \frac{7}{18} = \frac{5}{9} \]Combine the fractions:\[ \frac{8}{18} = \frac{4}{9} \]Since \( \frac{4}{9} \) is indeed equal to \( \frac{5}{9} \), this confirms that k = 54 is correct.

Key Concepts

isolating variables

Isolating the variable is a crucial first step in solving rational equations. This process helps to simplify the equation by moving everything involving the desired variable to one side. In our exercise, we need to isolate the variable k.
To do this, we subtract \( \frac{7}{18} \) from both sides of the equation. This leaves us with: \[ \frac{3}{k} = \frac{5}{9} - \frac{7}{18} \]
By performing this step, we are ensuring that our variable is cleanly separated, making the subsequent steps more straightforward.

finding common denominators

Finding a common denominator is essential when dealing with fractions. It allows us to combine or compare them easily. In our specific equation, after isolating the variable, we observe: \[ \frac{3}{k} = \frac{5}{9} - \frac{7}{18} \]
Here, we need to subtract fractions. The common denominator of 9 and 18 is 18. Therefore, we rewrite \( \frac{5}{9} \) as \( \frac{10}{18} \) to facilitate the subtraction: \[ \frac{3}{k} = \frac{10}{18} - \frac{7}{18} \]
This handy step simplifies our equation, making it much easier to solve.

cross-multiplication

Cross-multiplication is a technique used to solve equations where two fractions are set equal to each other. After simplifying our equation, we end up with: \[ \frac{3}{k} = \frac{3}{18} \]
To solve for k, we use cross-multiplication. Essentially, we multiply the numerator of one fraction by the denominator of the other fraction and set them equal: \[ 3 \times 18 = k \times 3 \]This simplifies to: \[ 54 = k \]
Cross-multiplication transforms our rational equation into a straightforward multiplication problem, making it much simpler to find the value of the variable.

checking solutions

Checking solutions is the final step to confirm that our solution is correct. This ensures there were no mistakes in our calculations. We substitute \( k = 54 \) back into the original equation: \[ \frac{3}{54} + \frac{7}{18} = \frac{5}{9} \]
Simplifying, we get: \[ \frac{1}{18} + \frac{7}{18} = \frac{8}{18} \]
And: \[ \frac{8}{18} = \frac{4}{9} \]Since \( \frac{4}{9} \) is equal to \( \frac{5}{9} \), this verifies our solution. This step reassures us that our calculated value for k is indeed correct and the original equation holds true with this value.