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We are tasked with finding values of \( a \) and \( b \) so that the system of equations

\[
\begin{aligned}
3x + 6y &= 12 \\
x + ay &= b
\end{aligned}
\]

has **no solutions**.

### Step 1: Understanding the condition for no solutions

For a system of linear equations to have **no solution**, the lines represented by the equations must be **parallel**, meaning they have the same slope but different y-intercepts.

Let's begin by expressing both equations in slope-intercept form \( y = mx + c \), where \( m \) is the slope.

### Step 2: Find the slope of the first equation

Starting with the first equation \( 3x + 6y = 12 \), solve for \( y \):

\[
6y = -3x + 12
\]

\[
y = -\frac{1}{2}x + 2
\]

So, the slope of this line is \( m_1 = -\frac{1}{2} \).

### Step 3: Find the slope of the second equation

Now, consider the second equation \( x + ay = b \). Solve for \( y \):

\[
ay = -x + b
\]

\[
y = -\frac{1}{a}x + \frac{b}{a}
\]

So, the slope of this line is \( m_2 = -\frac{1}{a} \).

### Step 4: Set the slopes equal for parallel lines

For the lines to be parallel, the slopes must be equal:

\[
-\frac{1}{a} = -\frac{1}{2}
\]

This simplifies to:

\[
a = 2
\]

### Step 5: Determine the condition for no solutions

For the system to have no solutions, the lines must also have **different y-intercepts**. The y-intercept of the first equation is 2 (from \( y = -\frac{1}{2}x + 2 \)). The y-intercept of the second equation is \( \frac{b}{a} = \frac{b}{2} \).

For the lines to not intersect, we require:

\[
\frac{b}{2} \neq 2
\]

This implies:

\[
b \neq 4
\]

### Final Answer:
- \( a = 2 \)
- \( b \neq 4 \)

### Relationship between the graphs:
The graphs of the equations will be parallel because they have the same slope but different y-intercepts. This means the lines will never intersect, resulting in no solution for the system.

---

Would you like further details on any part of the solution?

Here are five related questions for further exploration:

1. How do you determine if a system of equations has infinite solutions?
2. What are the geometric implications of two lines having the same slope but different y-intercepts?
3. Can you derive a condition for when two lines are perpendicular?
4. How do systems of equations change when solved using matrices or determinants?
5. How does the substitution method compare with the elimination method for solving systems of equations?

**Tip:** When solving systems of equations, visualizing the graphs of the lines can help you understand the nature of the solutions (one solution, no solution, or infinite solutions).

Find values of a and b so that the system 3x + 6y = 12 and x + ay = b has no solutions. Explain how the graphs of the equations will be related.

Learn how to find values of a and b so that the system of linear equations has no solutions. This involves understanding the slopes of the lines and conditions for parallelism. Suitable for students in Grades 9-11.

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