Blog: `x + 2y = 0, x + y = 6, 3x - 2y = 8` Use matricies to solve the system of equations. Use Gaussian elimination with back-substitution.

Sunday, August 28, 2011

The augmented matrix $\displaystyle{\left[\matrix{{1}&{2}&{0}\\{1}&{1}&{6}\\{3}&-{2}&{8}}\right]}$

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

$\displaystyle{\left[\matrix{{1}&{2}&{0}\\{0}&{1}&-{6}\\{3}&-{2}&{8}}\right]}$

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

$\displaystyle{\left[\matrix{{1}&{2}&{0}\\{0}&{1}&-{6}\\{0}&{8}&-{8}}\right]}$

On applying $\displaystyle{R}_{{3}}-\gt\frac{{{R}_{{3}}}}{{8}}$ we get

$\displaystyle{\left[\matrix{{1}&{2}&{0}\\{0}&{1}&-{6}\\{0}&{1}&-{1}}\right]}$

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

$\displaystyle{\left[\matrix{{1}&{2}&{0}\\{0}&{1}&-{6}\\{0}&{0}&{5}}\right]}$

The corresponding system is $\displaystyle{x}+{2}{y}={0}$

$\displaystyle{y}=-{6}$

$\displaystyle{0}{\left({x}\right)}+{0}{\left({y}\right)}={5}$

No $\displaystyle{x}$ and $\displaystyle{y}$ value satisfy the last equation.

Hence the system is inconsistent.

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