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Chapter 8: Problem 20

In the following exercises, solve each equation. $$ p-\frac{2}{5}=\frac{2}{3} $$

Short Answer

Expert verified
p = \frac{16}{15}

Step by step solution

01

- Isolate the variable term

Add \(\frac{2}{5}\) to both sides of the equation. This gives: \[ p - \frac{2}{5} + \frac{2}{5} = \frac{2}{3} + \frac{2}{5} \] Simplify this to: \[ p = \frac{2}{3} + \frac{2}{5} \]
02

- Find a common denominator

To add the fractions \(\frac{2}{3}\) and \(\frac{2}{5}\), find a common denominator. The lowest common denominator for 3 and 5 is 15: \[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \] \[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \]
03

- Add the fractions

Now add the fractions with the common denominator: \[ p = \frac{10}{15} + \frac{6}{15} = \frac{16}{15} \]
04

- Simplify if necessary

The fraction \(\frac{16}{15}\) is already in its simplest form. Therefore, the solution is: \[ p = \frac{16}{15} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isolate the variable
When solving linear equations, the first step is often to isolate the variable. This means getting the variable you are solving for on one side of the equation by itself.

In our example, the original equation is: \[ p - \frac{2}{5} = \frac{2}{3} \]
To isolate \text{p}\b, we need to get rid of \frac{2}{5}\b that is subtracted from it.

  • We do this by adding \frac{2}{5}\b to both sides of the equation.

After adding \frac{2}{5}\b, the equation becomes: \[ p = \frac{2}{3} + \frac{2}{5} \]

Once the variable is isolated, we can then proceed to the next steps to find its value.
common denominator
When adding fractions, you need a common denominator so that the fractions can be combined easily.

In our example, after isolating the variable, we have: \[ p = \frac{2}{3} + \frac{2}{5} \]
To add \frac{2}{3}\b and \frac{2}{5}\b, we first find their common denominator.

  • The least common denominator (LCD) for 3 and 5 is 15.
  • Convert each fraction to have 15 as the denominator.
  • For \frac{2}{3}\b: \[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]
  • For \frac{2}{5}\b: \[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \]

Now, both fractions have a common denominator of 15.
simplifying fractions
After getting a common denominator, the next step is to add the fractions and simplify if necessary.

In our example, we have \[ p = \frac{10}{15} + \frac{6}{15} \]

  • Add the numerators: \[ \frac{10}{15} + \frac{6}{15} = \frac{10 + 6}{15} = \frac{16}{15} \]

Now we have \frac{16}{15}\b.
Check if the fraction can be simplified further.
  • A fraction is simplest when the numerator and denominator have no common factors other than 1.
As \frac{16}{15}\b already has no common factors, it is in its simplest form. Thus, the solution to the equation is: \[ p = \frac{16}{15} \]

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