Solve the following LPP graphically:aximize:$subject to th

Question

Solve the following LPP graphically:

aximize:

$$z = 10x_1 + 10x_2$ $
subject to the constraints:
$$4x_1 + 3x_2 \le 12$ $
$$6x_1 + 18x_2 \le 36$ $
$$x_1, x_2 \ge 0.$ $
b) Find all values of k for which the vectors:
$$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix} 2 \\ -k \\ 2k \end{bmatrix}$ $
are linearly independent.

09 Jan 2026

Answer :

Word Count : 509

The given Linear Programming Problem (LPP) is: Maximize ( z = 10x_1 + 10x_2 ) subject to: [ 4x_1 + 3x_2 \le 12, \quad 6x_1 + 18x_2 \le 36, \quad x_1, x_2 \ge 0 ] First, we convert the inequalities to equalities to find the boundary lines: 1. ( 4x_1 + 3x_2 = 12 ) * (x_1)-intercept: (x_2 = 0 \Rightarrow 4x_1 = 12 \Rightarrow x_1 = 3) * (x_2)-intercept: (x_1 = 0 \Rightarrow 3x_2 = 12 \Rightarrow x_2 = 4) 2. ( 6x_1 + 18x_2 = 36 ) * (x_1)-intercept: (x_2 = 0 \Rightarrow 6x_1 = 36 \Rightarrow x_1 = 6) * (x_2)-intercept: (x_1 = 0 \Rightarrow 18x_2 = 36 \Rightarrow x_2 = 2) Next, find the intersection of the two lines: [ 4x_1 + 3x_2 ____ ________ _____ ________ ____ _____ ____ _______.
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