Maximizing Variable 'a' with Constraints

The question asks for the maximum possible value of the variable $a$, given an equation and constraints on the other variables.

Given Equation and Constraints

We are given the equation:

$a + b + c = 8$

Additionally, we have the constraints that $b$ and $c$ must be positive integers. This means:

• $b \ge 1$
• $c \ge 1$
• $b, c \in \mathbb{Z} \text{ (integers)}$

Strategy for Maximizing 'a'

To find the maximum value of $a$, we need to rearrange the equation:

$a = 8 - (b + c)$

This shows that maximizing $a$ is equivalent to minimizing the sum $(b + c)$.

Determining Minimum Sum of 'b' and 'c'

Given the constraints $b \ge 1$ and $c \ge 1$, the smallest possible integer value for $b$ is 1, and the smallest possible integer value for $c$ is 1.

Therefore, the minimum value of the sum $(b + c)$ is:

$(b + c)_{\text{min}} = 1 + 1 = 2$

Calculating Maximum Value of 'a'

Now, substitute the minimum value of $(b + c)$ back into the equation for $a$ to find its maximum value:

$a_{\text{max}} = 8 - (b + c)_{\text{min}}$

$a_{\text{max}} = 8 - 2$

$a_{\text{max}} = 6$

The maximum value 'a' can take is 6. This corresponds to Option A.