`5x - 3y + 2z = 3, 2x + 4y - z = 2, x + y - z = -1` Solve the system of linear equations and check any solutions algebraically.

Thursday, December 1, 2016

$\displaystyle{5}{x}-{3}{y}+{2}{z}={3},{2}{x}+{4}{y}-{z}={2},{x}+{y}-{z}=-{1}$ Solve the system of linear equations and check any solutions algebraically.

Eq. 1 : $\displaystyle{5}{x}-{3}{y}+{2}{z}={3}$

Eq. 2 : $\displaystyle{2}{x}+{4}{y}-{z}={2}$

Eq. 3 : $\displaystyle{x}+{y}-{z}=-{1}$

Multiply Eq. 2 by 2 and add Eq. 1,

$\displaystyle{4}{x}+{8}{y}-{2}{z}={4}$

$\displaystyle{5}{x}-{3}{y}+{2}{z}={3}$

Eq.4 : $\displaystyle{9}{x}+{5}{y}={7}$

Subtract Eq. 3 from Eq. 2,

Eq.5 : $\displaystyle{x}+{3}{y}={3}$

Now let's solve Eq.4 and Eq.5 by substitution method,

From Eq.5 ,

$\displaystyle{x}={3}-{3}{y}$

Substitute the above expression of x in the Eq.4,

$\displaystyle{9}{\left({3}-{3}{y}\right)}+{5}{y}={7}$

$\displaystyle{27}-{27}{y}+{5}{y}={7}$

$\displaystyle{27}-{22}{y}={7}$

$\displaystyle-{22}{y}={7}-{27}$

$\displaystyle-{22}{y}=-{20}$

$\displaystyle{y}=\frac{{-{20}}}{{-{{22}}}}$

$\displaystyle{y}=\frac{{10}}{{11}}$

Plug in the value of y in the expression of x,

$\displaystyle{x}={3}-{3}{y}$

$\displaystyle{x}={3}-{3}{\left(\frac{{10}}{{11}}\right)}$

$\displaystyle{x}={3}-\frac{{30}}{{11}}$

$\displaystyle{x}=\frac{{{33}-{30}}}{{11}}$

$\displaystyle{x}=\frac{{3}}{{11}}$

Plug in the values of x and y in the Eq.3,

$\displaystyle\frac{{3}}{{11}}+\frac{{10}}{{11}}-{z}=-{1}$

$\displaystyle\frac{{13}}{{11}}-{z}=-{1}$

$\displaystyle-{z}=-{1}-\frac{{13}}{{11}}$

$\displaystyle-{z}=\frac{{-{11}-{13}}}{{11}}$

$\displaystyle-{z}=\frac{{-{24}}}{{11}}$

$\displaystyle{z}=\frac{{24}}{{11}}$

Solutions of the equations are x=3/11 , y=10/11 and z=24/11

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