https://file.aiquickdraw.com/mathbot/file/1722438968830ue0jnayn.jpg

Let's solve the simultaneous equations provided in the image:

**Equation Set (1):**
1. \( 2(x - 1) = 3(y + 2) \)
2. \( 2y + 3 = 2(x - y) \)

**Equation Set (2):**
1. \( \frac{x + 2}{3} = \frac{u}{4} \)
2. \( \frac{x}{5} = u + 3 \)

### Solution for Set (1)
1. **First Equation:**
   \[ 2(x - 1) = 3(y + 2) \]
   \[ 2x - 2 = 3y + 6 \]
   \[ 2x - 3y = 8 \quad \text{(Equation A)} \]

2. **Second Equation:**
   \[ 2y + 3 = 2(x - y) \]
   \[ 2y + 3 = 2x - 2y \]
   \[ 4y = 2x - 3 \]
   \[ 2x - 4y = 3 \quad \text{(Equation B)} \]

### Solving Set (1)
Let's solve Equations A and B simultaneously:
1. **Equation A:** \( 2x - 3y = 8 \)
2. **Equation B:** \( 2x - 4y = 3 \)

Subtract Equation B from Equation A:
\[ (2x - 3y) - (2x - 4y) = 8 - 3 \]
\[ y = 5 \]

Substitute \( y = 5 \) into Equation A:
\[ 2x - 3(5) = 8 \]
\[ 2x - 15 = 8 \]
\[ 2x = 23 \]
\[ x = 11.5 \]

**Solution for Set (1):** \( x = 11.5 \), \( y = 5 \)

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### Solution for Set (2)
1. **First Equation:**
   \[ \frac{x + 2}{3} = \frac{u}{4} \]
   Cross-multiply:
   \[ 4(x + 2) = 3u \]
   \[ 4x + 8 = 3u \quad \text{(Equation C)} \]

2. **Second Equation:**
   \[ \frac{x}{5} = u + 3 \]
   \[ x = 5u + 15 \quad \text{(Equation D)} \]

### Solving Set (2)
Substitute Equation D into Equation C:
\[ 4(5u + 15) + 8 = 3u \]
\[ 20u + 60 + 8 = 3u \]
\[ 20u - 3u = -68 \]
\[ 17u = -68 \]
\[ u = -4 \]

Substitute \( u = -4 \) into Equation D:
\[ x = 5(-4) + 15 \]
\[ x = -20 + 15 \]
\[ x = -5 \]

**Solution for Set (2):** \( x = -5 \), \( u = -4 \)

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Would you like a detailed explanation of any specific step? Here are some related questions you might explore:

1. How do you verify the solutions of simultaneous equations?
2. What are the methods to solve simultaneous equations?
3. What is the graphical interpretation of solving two equations simultaneously?
4. How do different coefficients affect the solution of simultaneous equations?
5. What happens when simultaneous equations have no solution?
6. How do you solve simultaneous equations using matrices?
7. What is the role of substitution in solving simultaneous equations?
8. How can you use technology to solve simultaneous equations?

**Tip:** Always check your solutions by substituting them back into the original equations to ensure they satisfy all given conditions.

Learn how to solve simultaneous equations with detailed step-by-step solutions. This problem covers key mathematical concepts including simultaneous linear equations and fractional equations. Suitable for Grades 9-12 students.

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