Blog: `x + 2y + z = 8, 3x + 7y + 6z = 26` Use matricies to solve the system of equations (if possible). Use Gauss-Jordan elimination.

Friday, December 25, 2015

$\displaystyle{x}+{2}{y}+{z}={8},{3}{x}+{7}{y}+{6}{z}={26}$ Use matricies to solve the system of equations (if possible). Use Gauss-Jordan elimination.

The augmented matrix is $\displaystyle{\left[\matrix{{1}&{2}&{1}&{8}\\{3}&{7}&{6}&{26}}\right]}$

On applying $\displaystyle{R}_{{2}}-\gt{R}_{{2}}-{3}{R}_{{1}}$ we get

$\displaystyle{\left[\matrix{{1}&{2}&{1}&{8}\\{0}&{1}&{3}&{2}}\right]}$

On applying $\displaystyle{R}_{{1}}-\gt{R}_{{1}}-{2}{R}_{{2}}$ we get

$\displaystyle{\left[\matrix{{1}&{0}&-{5}&{4}\\{0}&{1}&{3}&{2}}\right]}$

The corresponding system is $\displaystyle{x}-{5}{z}={4}$

$\displaystyle{y}+{3}{z}={2}$

Let $\displaystyle{k}$ is any real number and $\displaystyle{z}={k}$ , now substitute $\displaystyle{z}$ value in the above equations to find $\displaystyle{x}$ and $\displaystyle{y}$ values.

$\displaystyle{x}-{5}{z}={4}$

$\displaystyle{x}={4}+{5}{z}={4}+{5}{k}$

$\displaystyle{y}+{3}{z}={2}$

$\displaystyle{y}={2}-{3}{z}={2}-{3}{k}$

$\displaystyle\therefore$ the solution set is $\displaystyle{\left\lbrace{4}+{5}{k},{2}-{3}{k},{k}\right\rbrace}$

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Friday, December 25, 2015

$\displaystyle{x}+{2}{y}+{z}={8},{3}{x}+{7}{y}+{6}{z}={26}$ Use matricies to solve the system of equations (if possible). Use Gauss-Jordan elimination.

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