Problem 3 - Elementary Algebra

Question

For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{9} p-\frac{1}{8}=\frac{25}{72} $$

Step-by-Step Solution

Answer

p = \frac{17}{16}

Step 1: Eliminate the fraction

To simplify the equation, clear the fractions by finding the least common denominator (LCD) of 9, 8, and 72, which is 72. Multiply both sides of the equation by 72: \[ 72 \times \frac{4}{9} p - 72 \times \frac{1}{8} = 72 \times \frac{25}{72} \]

Step 2: Simplify the equation

Simplify each term: \[ 72 \times \frac{4}{9} = 32 \] and \[ 72 \times \frac{1}{8} = 9 \] and \[ 72 \times \frac{25}{72} = 25 \] Thus, the equation simplifies to: \[ 32p - 9 = 25 \]

Step 3: Solve for p

Add 9 to both sides of the equation to isolate the term containing p: \[ 32p - 9 + 9 = 25 + 9 \] \[ 32p = 34 \] Now, divide both sides by 32 to solve for p: \[ p = \frac{34}{32} = \frac{17}{16} \]

Step 4: Check the solution

To verify the solution, substitute \( p = \frac{17}{16} \) back into the original equation: \[ \frac{4}{9} \times \frac{17}{16} - \frac{1}{8} = \frac{25}{72} \] Simplify each part: \[ \frac{4 \times 17}{9 \times 16} = \frac{68}{144} = \frac{17}{36} \] and \[ \frac{1}{8} = \frac{9}{72}\] Thus: \[ \frac{17}{36} - \frac{9}{72} = \frac{34}{72} - \frac{9}{72} = \frac{25}{72} \] The original equation is satisfied, confirming the solution is correct.

Key Concepts

least common denominator

When solving linear equations with fractions, it's crucial to 'clear' the fractions to make the equation easier to work with. The first step is finding the Least Common Denominator (LCD). The LCD is the smallest number that all the denominators of the fractions share as a multiple. For the fractions in our exercise with denominators 9, 8, and 72, the LCD is 72. Using the LCD helps simplify the equation by eliminating the fractions, streamlining subsequent steps.

clearing fractions

Clearing fractions means eliminating the fractional terms by multiplying each term by the Least Common Denominator. In our given equation: \text{\( \frac{4}{9} p - \frac{1}{8} = \frac{25}{72} \)}, we multiply every term by 72 (the LCD):

• \text{\( 72 \times \frac{4}{9} p = 32p \)}
• \text{\( 72 \times \frac{1}{8} = 9 \)}
• \text{\( 72 \times \frac{25}{72} = 25 \)}

After this step, the equation becomes much simpler:\text{\( 32p - 9 = 25 \)}. Now, the equation no longer contains fractions, making the next steps more straightforward.

isolation of variables

Now that we have a simplified equation without fractions, the next step is to isolate the variable to find its value. Starting with the equation \text{\( 32p - 9 = 25 \)}, we need to isolate p. We do this by adding 9 to both sides:

• \text{\( 32p - 9 + 9 = 25 + 9 \)}

This simplifies to:\text{\( 32p = 34 \)}. Next, we divide both sides by 32 to solve for p:\text{\( p = \frac{34}{32} = \frac{17}{16} \)}. At this point, we've found the solution for the variable p.

equation verification

To ensure our solution is correct, we need to verify it by substituting p back into the original equation. Let's check our found solution \text{\( p = \frac{17}{16} \)}:

• \text{\( \frac{4}{9} \times \frac{17}{16} - \frac{1}{8} = \frac{25}{72} \)}
• Simplifying, \text{\( \frac{4 \times 17}{9 \times 16} = \frac{68}{144} = \frac{17}{36} \)}
• And \text{\( \frac{1}{8} = \frac{9}{72} \)}

Thus, \text{\( \frac{17}{36} - \frac{9}{72} = \frac{34}{72} - \frac{9}{72} = \frac{25}{72} \)}. Since the left side equals the right side of the original equation, we confirm that \text{\( p = \frac{17}{16} \)} is indeed the correct solution. Verification is a vital step to ensure accuracy and reliability in solving equations.