`4x + 3y + 17z = 0, 5x + 4y + 22z = 0, 4x + 2y + 19z =0` Solve the system of linear equations and check any solutions algebraically.

Saturday, February 6, 2010

You may subtract the third equation from the first, such that:

$\displaystyle{y}-{2}{z}={0}\Rightarrow{y}={2}{z}$

Replace 2z for y in equations 2 and 3, such that:

$\displaystyle{5}{x}+{8}{z}+{22}{z}={0}\Rightarrow{5}{x}+{30}{z}={0}\Rightarrow{x}+{6}{z}={0}\Rightarrow{x}=-{6}{z}$

$\displaystyle{4}{x}+{4}{z}+{19}{z}={0}\Rightarrow{4}{x}+{23}{z}={0}$

$\displaystyle{4}{x}+{23}{z}-{4}{x}-{24}{z}={0}\Rightarrow-{z}={0}\Rightarrow{z}={0}\Rightarrow{x}={0}\Rightarrow{y}={0}$

Hence, evaluating the solution to the homogeneous system, yields x = y = z = 0.

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