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Chapter 12: Problem 78

To prepare for Section 12.6, review solving equations. Solve. $$ \frac{5}{3}=2 t $$

Short Answer

Expert verified
The solution is \(t = \frac{5}{6}\).

Step by step solution

01

- Isolate the variable

To solve the equation \(\frac{5}{3} = 2t\), first isolate the variable \(t\). In this case, divide both sides of the equation by 2.
02

- Perform the division

Divide \(\frac{5}{3}\) by 2: \(\frac{5}{3} \times \frac{1}{2} = \frac{5}{6}\). This simplifies the equation to \(t = \frac{5}{6}\).
03

- State the solution

Now that \(t\) is isolated, the solution to the equation is \(t = \frac{5}{6}\).

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Key Concepts

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

Isolating Variables
Isolating variables is a foundational skill in algebra. It involves manipulating the equation so that the variable we are solving for stands alone on one side of the equation. In our problem, the goal is to isolate the variable 't'. Starting with the equation \(\frac{5}{3} = 2t\), we see that 't' is being multiplied by 2. To isolate 't', we need to undo this multiplication by performing the opposite operation: division. We divide both sides of the equation by 2, leading to \(\frac{5}{3} \times \frac{1}{2} = t\). Now, 't' is isolated.
Solving for t
Solving for a specific variable means finding its value. After isolating the variable 't' in our equation, we found that \(\frac{5}{3} \times \frac{1}{2} = t\). When we perform this calculation, we simplify it to find the value of 't'. This results in \(t = \frac{5}{6}\). This process underscores the importance of isolating the variable first, as it makes solving for the variable straightforward.
Fraction Multiplication
Fraction multiplication is simple once you understand the rules. When multiplying fractions, you multiply the numerators together and then the denominators together. In our problem, we multiplied \(\frac{5}{3}\) by \(\frac{1}{2}\). The multiplication looks like this: \[ \frac{5 \times 1}{3 \times 2} = \frac{5}{6} \] Therefore, \(\frac{5}{3} \times \frac{1}{2} = \frac{5}{6}\). Always remember to multiply across the top and then across the bottom, and simplify the fraction if possible.
Simplifying Equations
Simplifying equations involves making them easier to read and solve. This often includes combining like terms, eliminating fractions by finding common denominators, or performing basic arithmetic operations. In the given problem, after multiplying the fractions, we get \(\frac{5}{6} = t\). This is already in its simplest form, as there are no common factors between the numerator and the denominator. Always check your final solution to ensure it is fully simplified. This makes your answer clear and helps to verify your calculations.

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