Rational equations 7/x + 1/4 = 9/x , show your solution and find the LCD

We are given the rational equation:

\[
\frac{7}{x} + \frac{1}{4} = \frac{9}{x}
\]

### Step 1: Identify the Least Common Denominator (LCD)

To solve the equation, we need to find the Least Common Denominator (LCD) of all the denominators. The denominators in this equation are \(x\) and \(4\). The LCD is the smallest number that both denominators divide into evenly.

- The LCD of \(x\) and \(4\) is \(4x\).

### Step 2: Multiply the entire equation by the LCD

To eliminate the denominators, we multiply every term by the LCD, \(4x\):

\[
4x \cdot \left(\frac{7}{x}\right) + 4x \cdot \left(\frac{1}{4}\right) = 4x \cdot \left(\frac{9}{x}\right)
\]

### Step 3: Simplify each term

Now, simplify the terms:

\[
4 \cdot 7 + x \cdot 1 = 4 \cdot 9
\]

This simplifies to:

\[
28 + x = 36
\]

### Step 4: Solve for \(x\)

Now, isolate \(x\) by subtracting 28 from both sides:

\[
x = 36 - 28
\]

\[
x = 8
\]

### Final Answer:

The solution to the equation is \(x = 8\).

### Verification:

Substitute \(x = 8\) back into the original equation:

\[
\frac{7}{8} + \frac{1}{4} = \frac{9}{8}
\]

Convert \(\frac{1}{4}\) to a denominator of 8:

\[
\frac{1}{4} = \frac{2}{8}
\]

Now check if the equation holds:

\[
\frac{7}{8} + \frac{2}{8} = \frac{9}{8}
\]

Since \( \frac{9}{8} = \frac{9}{8} \), the solution \(x = 8\) is correct.

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Do you need more details or have any questions?

Here are five related questions:
1. How do you handle rational equations with more complex denominators?
2. What are the steps to solve a rational equation when the LCD involves variables and constants?
3. How can extraneous solutions arise when solving rational equations?
4. What happens if the variable \(x\) equals 0 in a rational equation?
5. How do you solve a rational equation when the LCD is a product of multiple variables?

**Tip:** Always check for extraneous solutions, especially when variables appear in the denominator.

Learn how to solve the rational equation 7/x + 1/4 = 9/x by finding the Least Common Denominator (LCD) and eliminating denominators. This step-by-step solution explains the use of algebraic principles to isolate the variable. Suitable for students in Grades 8-10.

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