https://file.aiquickdraw.com/mathbot/file/1729560848396irqjuso9.jpg

Let's solve a few of the equations from the image step by step.

### 35. \(\frac{x}{2} - \frac{x}{3} = 4\)

To solve this:
1. Find a common denominator on the left-hand side. The common denominator between 2 and 3 is 6.
   \[
   \frac{x}{2} = \frac{3x}{6}, \quad \frac{x}{3} = \frac{2x}{6}
   \]
   So the equation becomes:
   \[
   \frac{3x}{6} - \frac{2x}{6} = 4
   \]
2. Simplify the left-hand side:
   \[
   \frac{x}{6} = 4
   \]
3. Multiply both sides by 6:
   \[
   x = 24
   \]

So the solution to equation 35 is \(x = 24\).

---

### 36. \(\frac{x}{2} + \frac{x}{3} = 10\)

1. Find a common denominator for the left-hand side. Again, the common denominator between 2 and 3 is 6:
   \[
   \frac{x}{2} = \frac{3x}{6}, \quad \frac{x}{3} = \frac{2x}{6}
   \]
   So the equation becomes:
   \[
   \frac{3x}{6} + \frac{2x}{6} = 10
   \]
2. Simplify the left-hand side:
   \[
   \frac{5x}{6} = 10
   \]
3. Multiply both sides by 6:
   \[
   5x = 60
   \]
4. Divide by 5:
   \[
   x = 12
   \]

So the solution to equation 36 is \(x = 12\).

---

### 37. \(\frac{x}{6} + 1 = \frac{x}{3}\)

1. First, eliminate the fractions by multiplying through by 6:
   \[
   x + 6 = 2x
   \]
2. Subtract \(x\) from both sides:
   \[
   6 = x
   \]

So the solution to equation 37 is \(x = 6\).

---

### 38. \(\frac{3}{2}(y + 4) = \frac{20 - y}{2}\)

1. Multiply both sides of the equation by 2 to eliminate the fractions:
   \[
   3(y + 4) = 20 - y
   \]
2. Distribute on the left-hand side:
   \[
   3y + 12 = 20 - y
   \]
3. Add \(y\) to both sides:
   \[
   4y + 12 = 20
   \]
4. Subtract 12 from both sides:
   \[
   4y = 8
   \]
5. Divide by 4:
   \[
   y = 2
   \]

So the solution to equation 38 is \(y = 2\).

Would you like to go through the remaining problems or explore any of the steps in more detail? 

Here are 5 relative questions based on this:

1. What happens if both sides of an equation involve different denominators?
2. How can you simplify equations that involve fractional coefficients?
3. Why do we multiply through by the least common denominator when solving equations with fractions?
4. How do you check if your solution to an equation is correct?
5. What should you do if both sides of an equation cancel out to leave the same number?

**Tip:** When solving equations with fractions, always try to eliminate the denominators early to simplify the process.

Solve each equation from the provided set, including equations with fractions.

Learn how to solve algebraic equations involving fractions, including finding common denominators and simplifying both sides. Covers key algebraic concepts such as solving linear equations with fractions and properties of equality. Suitable for students in Grades 8-10.

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