`x - 3y + 2z = 18, 5x - 13y + 12z = 80` Solve the system of linear equations and check any solutions algebraically.

Thursday, June 16, 2016

You should notice that the system has a smaller number of equations than the number of variables, hence, the system is indeterminate.

$\displaystyle{x}={3}{y}-{2}{z}+{18}$

Replace $\displaystyle{3}{y}-{2}{z}+{18}$ for x in the second equation, such that:

$\displaystyle{5}{\left({3}{y}-{2}{z}+{18}\right)}-{13}{y}+{12}{z}={80}\Rightarrow{15}{y}-{10}{z}+{90}-{13}{y}+{12}{z}={80}$

$\displaystyle{2}{y}+{2}{z}=-{10}\Rightarrow{y}+{z}=-{5}\Rightarrow{y}=-{5}-{z}\Rightarrow{x}={3}{\left(-{5}-{z}\right)}-{2}{z}+{18}$

$\displaystyle{x}={3}-{5}{z}$

Hence, evaluating the solutions to the system yields $\displaystyle{x}={3}-{5}{z},{y}=-{5}-{z},{z}={z}.$

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