https://r2.erweima.ai/mathbot/image/1725440527243f67o0gbj.jpg

The image presents four different algebraic equations. Each equation involves a variable \( x \) and fractions where \( x \) is in the denominator. Let's solve each equation individually to find the value of \( x \).

### Equations:
1. \( 4 + \frac{1}{x} = \frac{8}{x} \)
2. \( 3 + \frac{1}{x} = \frac{7}{x} \)
3. \( 2 + \frac{1}{x} = \frac{5}{x} \)

### Solving the Equations:

1. **Equation 1:**
   \[
   4 + \frac{1}{x} = \frac{8}{x}
   \]
   Subtract \( \frac{1}{x} \) from both sides:
   \[
   4 = \frac{7}{x}
   \]
   Multiply both sides by \( x \):
   \[
   4x = 7
   \]
   Divide both sides by 4:
   \[
   x = \frac{7}{4}
   \]
   So, \( x = 1.75 \).

2. **Equation 2:**
   \[
   3 + \frac{1}{x} = \frac{7}{x}
   \]
   Subtract \( \frac{1}{x} \) from both sides:
   \[
   3 = \frac{6}{x}
   \]
   Multiply both sides by \( x \):
   \[
   3x = 6
   \]
   Divide both sides by 3:
   \[
   x = 2
   \]

3. **Equation 3:**
   \[
   2 + \frac{1}{x} = \frac{5}{x}
   \]
   Subtract \( \frac{1}{x} \) from both sides:
   \[
   2 = \frac{4}{x}
   \]
   Multiply both sides by \( x \):
   \[
   2x = 4
   \]
   Divide both sides by 2:
   \[
   x = 2
   \]

### Summary of Results:
- From the first equation: \( x = 1.75 \)
- From the second equation: \( x = 2 \)
- From the third equation: \( x = 2 \)

It seems that the correct value of \( x \) that satisfies most equations here is \( x = 2 \).

Would you like more details on these calculations or have any other questions?

### Related Questions:
1. How do you solve linear equations with fractions?
2. What are some tips for solving algebraic equations with variables in the denominator?
3. Can you explain how to check solutions for algebraic equations?
4. What are other methods for solving equations involving fractions?
5. What are the common pitfalls to avoid when solving such algebraic equations?

### Tip:
Always simplify the equation as much as possible before trying to solve it; this can help avoid mistakes and make the solution more straightforward.

Solve the algebraic equations involving fractions and a variable x.

This guide walks through solving algebraic equations with fractions where the variable x is in the denominator. Learn the steps to isolate x and find its value using straightforward algebraic techniques. Suitable for students in Grades 7-9.

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