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Solving rationale equation using cross multiplication

The equation in the image is:

\[
\frac{2}{3} - \frac{1}{5} = \frac{7}{3x}
\]

We will solve this using cross multiplication. Let's break it down step by step.

### Step 1: Simplify the left-hand side (LHS)

First, we simplify the fractions on the left side of the equation:

\[
\frac{2}{3} - \frac{1}{5}
\]

To subtract these fractions, we need a common denominator. The least common denominator (LCD) between 3 and 5 is 15. Now we rewrite the fractions with this common denominator:

\[
\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}, \quad \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}
\]

So:

\[
\frac{2}{3} - \frac{1}{5} = \frac{10}{15} - \frac{3}{15} = \frac{7}{15}
\]

### Step 2: Set up the equation for cross multiplication

Now, the equation becomes:

\[
\frac{7}{15} = \frac{7}{3x}
\]

We can now apply cross multiplication. Cross-multiply to eliminate the denominators:

\[
7 \times 3x = 7 \times 15
\]

This simplifies to:

\[
21x = 105
\]

### Step 3: Solve for \( x \)

Now, solve for \( x \) by dividing both sides of the equation by 21:

\[
x = \frac{105}{21} = 5
\]

Thus, the solution is:

\[
x = 5
\]

Would you like any further clarification or details on the steps? Here are 5 related questions that build on this concept:

1. How do you find the least common denominator for more complex fractions?
2. What is the general process for solving rational equations with variables in the denominator?
3. Can cross multiplication be applied in equations where the fractions have more than one term in the numerator?
4. How would you approach solving this equation if the right-hand side had been more complicated?
5. Why is it necessary to check for restrictions (like \(x \neq 0\)) in rational equations?

**Tip**: Always double-check for extraneous solutions when dealing with rational equations by substituting your result back into the original equation.

Solving the equation (2/3) - (1/5) = 7/(3x) using cross multiplication.

This guide explains how to solve the rational equation (2/3) - (1/5) = 7/(3x) using cross multiplication, simplifying the fractions and solving for x. Key algebraic concepts such as finding the least common denominator and applying cross multiplication are covered. Suitable for students in Grades 8-10.

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