CUET UG 2025 — Mathematics Algebra

The maximum value of $z = 5x + 7y$z=5x+7y is found subject to the constraints $x + y \leq 5$x+y≤5, $x \geq 0$x≥0, $y \geq 0$y≥0.

In linear programming, the maximum or minimum value occurs at a corner point (vertex) of the feasible region.

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The corner points are where the boundary lines meet.

When $x = 0$x=0 and $y = 0$y=0: $(0, 0)$(0,0)

When $y = 0$y=0 and $x + y = 5$x+y=5, then $x = 5$x=5: $(5, 0)$(5,0)

When $x = 0$x=0 and $x + y = 5$x+y=5, then $y = 5$y=5: $(0, 5)$(0,5)

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Evaluating $z = 5x + 7y$z=5x+7y at each corner point:

At $(0, 0)$(0,0):

$$z = 5(0) + 7(0)$$z=5(0)+7(0)

$$z = 0$$z=0

At $(5, 0)$(5,0):

$$z = 5(5) + 7(0)$$z=5(5)+7(0)

$$z = 25$$z=25

At $(0, 5)$(0,5):

$$z = 5(0) + 7(5)$$z=5(0)+7(5)

$$z = 35$$z=35

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Comparing the values: $0$0, $25$25, and $35$35

The maximum value is $35$35, which occurs at the point $(0, 5)$(0,5).