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

Monday, March 31, 2008

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

From the third equation $\displaystyle{z}={x}+{y}+{1}.$ Substitute it into the first two equations and obtain

$\displaystyle{2}{x}+{4}{y}+{x}+{y}+{1}={1},$ or $\displaystyle{3}{x}+{5}{y}={0},$

and

$\displaystyle{x}-{2}{y}-{3}{x}-{3}{y}-{3}={2},$ or $\displaystyle{2}{x}+{5}{y}=-{5}.$

Now subtract these two equations to eliminate $\displaystyle{y}:$

$\displaystyle{x}={5}.$

Now $\displaystyle{y}=-\frac{{3}}{{5}}{x}=-{3}$ and $\displaystyle{z}={x}+{y}+{1}={3}.$

The answer: $\displaystyle{x}={5},$ $\displaystyle{y}=-{3},$ $\displaystyle{z}={3}.$ Check the solution:

$\displaystyle{2}{x}+{4}{y}+{z}={10}-{12}+{3}={1}$ (true),
$\displaystyle{x}-{2}{y}-{3}{z}={5}+{6}-{9}={2}$ (true),
$\displaystyle{x}+{y}-{z}={5}-{3}-{3}=-{1}$ (true).

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Monday, March 31, 2008

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

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