Obtain all the basic solutions to the following system of linear equat

Question

Obtain all the basic solutions to the following system of linear equations:

$$x_1 + 2x_2 + x_3 = 4$
$$2x_1 + x_2 + 5x_3 = 5$
Which of the solutions are feasible? Justify.

09 Jan 2026

Answer :

Word Count : 433

We are asked to find all basic solutions for the system: [ x_1 + 2x_2 + x_3 = 4 ] [ 2x_1 + x_2 + 5x_3 = 5 ] A basic solution is obtained by setting (n-m = 1) variable(s) to 0 (here, (n=3) variables, (m=2) equations → 1 variable to 0) and solving for the others. --- Step 1: Set (x_1 = 0) [ 0 + 2x_2 + x_3 = 4 \implies 2x_2 + x_3 = 4 ] [ 0 + x_2 + 5x_3 = 5 \implies x_2 + 5x_3 = 5 ] Solve the second equation for (x_2): [ x_2 = 5 - 5x_3 ] Substitute _____ _________ ________ ____ _________ ______ _____ ________ ____ ______ _____ _____.
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