If the objective function z = 4x + 3y has maximum value on a line joining points (3, a) and (b, 2) where a > 0, b > 0 such that a - b = 2, then the maximum value of z is:

The objective function is $z = 4x + 3y$ with maximum value occurring on a line joining points $(3, a)$ and $(b, 2)$.

Given conditions: $a > 0$, $b > 0$, and $a - b = 2$

In linear programming, if the maximum value occurs along an entire line segment rather than at a single point, the objective function has the same value at both endpoints. This happens when the line segment is parallel to the contour lines of the objective function.

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The objective function $z = 4x + 3y$ can be rewritten as:

$y = -\dfrac{4x}{3} + \dfrac{z}{3}$

The slope of contour lines (where z is constant) is $-\dfrac{4}{3}$

For the maximum to occur along the entire line from $(3, a)$ to $(b, 2)$, this line must have the same slope:

$\dfrac{2 - a}{b - 3} = -\dfrac{4}{3}$

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Cross-multiplying:

$3(2 - a) = -4(b - 3)$

$6 - 3a = -4b + 12$

$-3a + 4b = 6$

$4b - 3a = 6$ ... (Equation 1)

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From the given condition $a - b = 2$:

$a = b + 2$ ... (Equation 2)

Substituting Equation 2 into Equation 1:

$4b - 3(b + 2) = 6$

$4b - 3b - 6 = 6$

$b = 12$

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Using $a = b + 2$:

$a = 12 + 2$

$a = 14$

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The two points are $(3, 14)$ and $(12, 2)$

At point $(3, 14)$:

$z = 4(3) + 3(14)$

$z = 12 + 42$

$z = 54$

At point $(12, 2)$:

$z = 4(12) + 3(2)$

$z = 48 + 6$

$z = 54$

Therefore, the maximum value of z is $54$.