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Chapter 2: Problem 35

Solve. \(\frac{5}{3} \cdot y=\frac{10}{3}\)

Short Answer

Expert verified
y = 2

Step by step solution

01

Isolate the variable

To isolate the variable, divide both sides of the equation by \ \( \frac{5}{3} \) to get \ \( y \). This leads to \ \( y = \frac{10}{3} \div \frac{5}{3} \).
02

Simplify the equation

Perform the division on the right-hand side. Dividing fractions involves multiplying by the reciprocal. Therefore: \[ y = \frac{10}{3} \times \frac{3}{5} \]
03

Calculate the result

Simplify the multiplication: \ \( y = \frac{10 \times 3}{3 \times 5} = \frac{30}{15} \). Then, simplify: \[ y = 2 \]

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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 an equation, the goal is to get the variable on one side of the equation by itself. In our example, we started with \(\frac{5}{3} \cdot y = \frac{10}{3}\). To isolate the variable \(y\), we need to remove the fraction \(\frac{5}{3}\) that's being multiplied by \(y\). This is done by dividing both sides of the equation by \(\frac{5}{3}\).
When you divide both sides by \(\frac{5}{3}\), the equation becomes:
\(\frac{\frac{10}{3}}{\frac{5}{3}}\).
Dividing by a fraction is the same as multiplying by its reciprocal, so the next step involves.
Divide Fractions
Dividing fractions might seem intimidating, but it's really quite straightforward once you understand the process. First, recall that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator.
In our example, we need to divide \(\frac{10}{3}\) by \(\frac{5}{3}\). This becomes:
\frac{10}{3}\ \div\ \frac{5}{3}\ = \frac{10}{3} \; * \; \frac{3}{5}\.
Notice how we've replaced the division with multiplication by the reciprocal.
Now, we move on to simplifying the expression.
Simplify Expressions
Simplifying expressions involves reducing them to their simplest form. Here, we perform the multiplication of fractions. Multiply the numerators together and the denominators together:
\frac{10}{3}\ \cdot\ \frac{3}{5}\ = \frac{10 \times 3}{3 \times 5}\.
This simplifies to:
= \frac{30}{15}\.
We can further simplify \(\frac{30}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, in this case, 15:
\frac{30}{15}\ = 2.
Thus, the value of \ y\ is found to be 2!

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