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?

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.