Mathematics

Mathematics

Problem

Examine the following system of linear equations and solve it, if possible: x+y+z=6x + y + z = 6, 2x−y+3z=142x - y + 3z = 14, 3x+4y−2z=23x + 4y - 2z = 2. a. (i) Form the matrices AA, XX, and BB such that AX=BAX = B. b. (ii) Compute ∣A∣|A|. c. (iii) Using the value of ∣A∣|A|, comment on the nature of the solution of the system. d. (iv) If the system is consistent with a unique solution, find the values of xx, yy, and zz.
Intuitive explanation

The key to solving a system of three equations in three unknowns is elimination: strategically combine pairs of equations to cancel variables and reduce the system to simpler forms. Once you've eliminated one variable, you'll have two equations in two unknowns, which you can solve using the same technique. The final step is back-substitution to find all three values. Check whether the system is consistent (has a solution) or inconsistent by watching for contradictions as you simplify.

Intuitive explanation

The key to solving a system of three equations in three unknowns is elimination: strategically combine pairs of equations to cancel variables and reduce the system to simpler forms. Once you've eliminated one variable, you'll have two equations in two unknowns, which you can solve using the same technique. The final step is back-substitution to find all three values. Check whether the system is consistent (has a solution) or inconsistent by watching for contradictions as you simplify.

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