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Let's go through the questions step by step. The first question (#15) involves solving a system of linear equations. Let’s start by addressing this one.

### Problem 15:
#### The system of linear equations:
\[
x + 3y = 6 \quad \text{(Equation 1)}
\]
\[
2x - 3y = 12 \quad \text{(Equation 2)}
\]

We need to solve this system of equations, and we can do this by either substitution or elimination. Let’s use elimination.

1. **Add Equation 1 and Equation 2:**
\[
(x + 3y) + (2x - 3y) = 6 + 12
\]
This simplifies to:
\[
3x = 18 \implies x = 6
\]

2. **Substitute \( x = 6 \) into Equation 1:**
\[
6 + 3y = 6 \implies 3y = 0 \implies y = 0
\]

Thus, the solution is \( x = 6 \) and \( y = 0 \).

### Next Parts:
- **(a)** The point of intersection is the solution of the system, which is \( (6, 0) \).
- **(b)** The number of solutions of the equation \( x + 24 = 8 \) is straightforward:
   \[
   x + 24 = 8 \implies x = 8 - 24 = -16
   \]
   Hence, the solution is \( x = -16 \), and since it's a linear equation, there is **one solution**.

   For \( 2x + 4y = 16 \), the number of solutions depends on whether the system is consistent. This is also a line, so if paired with another independent equation, there is exactly **one solution** unless they are parallel, in which case no solution, or coincident, in which case infinitely many solutions exist.

- **(c)** If \( (1, 2) \) is the solution to the system \( ax + by = 3 \) and \( 2x + by = 12 \), we substitute \( x = 1 \) and \( y = 2 \) into the equations to find \( a \) and \( b \).

    1. From \( a(1) + b(2) = 3 \), we get:
       \[
       a + 2b = 3 \quad \text{(Equation 1)}
       \]
    2. From \( 2(1) + b(2) = 12 \), we get:
       \[
       2 + 2b = 12 \implies 2b = 10 \implies b = 5
       \]
    3. Substitute \( b = 5 \) into Equation 1:
       \[
       a + 2(5) = 3 \implies a + 10 = 3 \implies a = -7
       \]
    So, \( a = -7 \) and \( b = 5 \).

Would you like to continue solving the other questions, or do you need more details for this solution?

15. The system of linear equations: x + 3y = 6 and 2x - 3y = 12. (a) What is the point of intersection? (b) Find the number of solutions for x + 24 = 8 and 2x + 4y = 16. (c) If (1, 2) is the solution of ax + by = 3 and 2x + by = 12, find a and b.

Explore how to solve a system of linear equations, find points of intersection, and determine the number of solutions to linear equations. This detailed guide covers concepts of algebra and systems of equations, suitable for students in Grades 10-12.

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