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To solve the system of linear equations:

\[
x + 5y = b
\]
\[
ay = 6 + 2x
\]

and determine the value of \((a + b)^2\) when the system has infinitely many solutions, we can follow these steps:

### Step 1: Express the system in standard form
First, rewrite both equations in standard form:

1. \(x + 5y = b\)
2. \(ay = 2x + 6\), which can be rearranged as \(2x - ay = -6\).

Now, the system is:

\[
x + 5y = b
\]
\[
2x - ay = -6
\]

### Step 2: Condition for infinitely many solutions
For a system of linear equations to have infinitely many solutions, the two equations must represent the same line. This means the coefficients of \(x\), \(y\), and the constants on the right-hand side of both equations must be proportional.

From the first equation, the coefficients are:
- Coefficient of \(x\): \(1\)
- Coefficient of \(y\): \(5\)
- Constant on the right-hand side: \(b\)

From the second equation, the coefficients are:
- Coefficient of \(x\): \(2\)
- Coefficient of \(y\): \(-a\)
- Constant on the right-hand side: \(-6\)

Now, set up the proportionality condition:

\[
\frac{1}{2} = \frac{5}{-a} = \frac{b}{-6}
\]

### Step 3: Solve the proportions
From \(\frac{1}{2} = \frac{5}{-a}\):

\[
\frac{1}{2} = \frac{5}{-a} \quad \Rightarrow \quad a = -10
\]

From \(\frac{1}{2} = \frac{b}{-6}\):

\[
\frac{1}{2} = \frac{b}{-6} \quad \Rightarrow \quad b = -3
\]

### Step 4: Find \((a + b)^2\)
Now that we have \(a = -10\) and \(b = -3\), we can find \((a + b)^2\):

\[
a + b = -10 + (-3) = -13
\]
\[
(a + b)^2 = (-13)^2 = 169
\]

### Final Answer:
The value of \((a + b)^2\) is \(169\).

Would you like more details or have any other questions?

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Here are 5 questions that expand on this:

1. What is the geometric interpretation of a system having infinitely many solutions?
2. How do you find the condition for two lines to be identical in a system of linear equations?
3. What happens if two lines in a system of equations are parallel but not the same?
4. How can you solve a system of linear equations with substitution or elimination methods?
5. How would the solution change if the system had no solutions instead of infinitely many?

**Tip**: Always check the proportionality of coefficients when looking for infinitely many solutions in a system of linear equations.

In the system of linear equations, x + 5y = b and ay = 6 + 2x, where a and b are constants, if the system has infinitely many solutions, what is the value of (a + b)^2?

Learn how to solve the system of linear equations x + 5y = b and ay = 6 + 2x when it has infinitely many solutions. Find the value of (a + b)^2 using the proportionality condition for linear equations. Suitable for students in Grades 9-12.

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