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Chapter 3: Problem 53

Solve. $$ \frac{2}{3}+t=\frac{4}{5} $$

Short Answer

Expert verified
t = \frac{2}{15}

Step by step solution

01

Isolate the variable

To solve for the variable, isolate the variable term by subtracting \frac{2}{3} from both sides of the equation. \[ t = \frac{4}{5} - \frac{2}{3} \]
02

Find a common denominator

To subtract the fractions, find a common denominator. The least common multiple (LCM) of 5 and 3 is 15. Convert the fractions: \[ \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \] \[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]
03

Perform the subtraction

Now that the fractions have a common denominator, subtract them: \[ t = \frac{12}{15} - \frac{10}{15} = \frac{2}{15} \]

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Key Concepts

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

Common Denominator
When working with fractions, finding a common denominator is crucial. This allows us to operate on the fractions easily since they share the same base. To find the common denominator, you should determine the least common multiple (LCM) of the denominators. In our example, the denominators are 5 and 3. The LCM of 5 and 3 is 15. This means both fractions can be transformed so that they have the same denominator, making addition and subtraction straightforward.

Here's how you can convert the fractions:
  • First fraction: \( \frac{4}{5} \). Convert to \( \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \).
  • Second fraction: \( \frac{2}{3} \). Convert to \( \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \).
Now both fractions have a common denominator of 15, making them easier to manage.
Subtraction of Fractions
Subtraction of fractions becomes simple once they have a common denominator. You can subtract the numerators directly while keeping the denominator the same.

Consider our example:
After finding a common denominator, we have:
  • First fraction: \( \frac{12}{15} \)
  • Second fraction: \( \frac{10}{15} \)
Subtract the numerators: \(12 - 10 \) which results in 2. The denominator remains 15.
Thus, \( \frac{12}{15} - \frac{10}{15} = \frac{2}{15} \).

So, the result of the subtraction is \( \frac{2}{15} \). This method ensures accuracy and keeps the operation simple.
Isolating Variables
One of the fundamental steps in solving linear equations is isolating the variable. This means getting the variable on one side of the equation and all other terms on the opposite side. To isolate the variable, we manipulate the equation using basic arithmetic operations.

In our example:
The equation is \( \frac{2}{3} + t = \frac{4}{5} \).
  • Subtract \( \frac{2}{3} \) from both sides to get: \( t = \frac{4}{5} - \frac{2}{3} \)
Now, it's easier to solve for \( t \). After finding the common denominator and performing the subtraction, we found that \( t = \frac{2}{15} \).

Isolating the variable simplifies the problem and helps us see the solution clearly.

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