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Chapter 2: Problem 60

Solve using the multiplication principle. Don't forget to check! $$ \frac{2}{7}=\frac{x}{3} $$

Short Answer

Expert verified
\( x = \frac{6}{7} \)

Step by step solution

01

- Understand the equation

The given equation is \[ \frac{2}{7}=\frac{x}{3} \]. We need to solve for the variable \( x \).
02

- Use the multiplication principle

To isolate \( x \), we can cross-multiply. Cross-multiplying the two fractions gives: \[ 2 \times 3 = 7 \times x \].
03

- Simplify the equation

Simplify the equation obtained from cross-multiplying: \[ 6 = 7x \].
04

- Solve for \( x \)

Isolate \( x \) by dividing both sides of the equation by 7: \[ x = \frac{6}{7} \].
05

- Check the solution

Substitute \( x = \frac{6}{7} \) back into the original equation to verify: \[ \frac{2}{7} = \frac{\frac{6}{7}}{3} \] Simplify the right-hand side: \[ \frac{6}{7 \times 3} = \frac{6}{21} = \frac{2}{7} \] Both sides are equal, so the solution is correct.

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Key Concepts

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

cross-multiplication
When solving equations involving fractions, cross-multiplication is a powerful tool. It helps to eliminate the fractions by converting the equation into a simpler form. In the given exercise, the equation is \(\frac{2}{7} = \frac{x}{3}\). To apply cross-multiplication, you multiply the numerator of one fraction by the denominator of the other fraction. This step transforms our equation: \[2 \times 3 = 7 \times x\].
This gives us an equation without fractions, making it easier to solve for the variable.
solving equations
Solving equations means finding the value of the variable that makes the equation true. Once we have cross-multiplied and simplified the equation to \[6 = 7x\], our next goal is to isolate \(x\). To do this, we divide both sides of the equation by 7: \[x = \frac{6}{7}\].
This isolates \(x\), giving us the solution \(\frac{6}{7}\).
Remember, each step in solving the equation should be performed on both sides to maintain the equality.
checking solutions
After finding a solution, it's crucial to verify its correctness. To check our solution \(x = \frac{6}{7}\), we substitute it back into the original equation: \(\frac{2}{7} = \frac{\frac{6}{7}}{3}\).
Simplifying the right side: \[ \frac{\frac{6}{7}}{3} = \frac{6}{21} = \frac{2}{7} \],
shows that both sides of the equation are equal.
Hence, our solution is correct. This step ensures that the solution satisfies the original equation, confirming its validity.

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