Blog: `2x + 3y =0, 4x + 3y - z = 0, 8x + 3y + 3z = 0` Solve the system of linear equations and check any solutions algebraically.

Saturday, February 18, 2017

$\displaystyle{2}{x}+{3}{y}={0},{4}{x}+{3}{y}-{z}={0},{8}{x}+{3}{y}+{3}{z}={0}$ Solve the system of linear equations and check any solutions algebraically.

Eq1: $\displaystyle{2}{x}+{3}{y}={0}$

Eq2: $\displaystyle{4}{x}+{3}{y}-{z}={0}$

Eq3: $\displaystyle{8}{x}+{3}{y}+{3}{z}={0}$

Multiply equation 2 by 3,

$\displaystyle{12}{x}+{9}{y}-{3}{z}={0}$

Now add the above equation and equation 3,

$\displaystyle{\left({8}{x}+{3}{y}+{3}{z}\right)}+{\left({12}{x}+{9}{y}-{3}{z}\right)}={0}$

$\displaystyle{8}{x}+{12}{x}+{3}{y}+{9}{y}+{3}{z}-{3}{z}={0}$

$\displaystyle{20}{x}+{12}{y}={0}$

$\displaystyle{4}{\left({5}{x}+{3}{y}\right)}={0}$

$\displaystyle{5}{x}+{3}{y}={0}$

Now let's solve the above equation and equation 1,

$\displaystyle{2}{x}+{3}{y}={0}$

$\displaystyle{5}{x}+{3}{y}={0}$

Subtract the above equations

$\displaystyle{2}{x}-{5}{x}={0}$

$\displaystyle-{3}{x}={0}$

$\displaystyle{x}={0}$

Substitute back the value of x,

$\displaystyle{2}{x}+{3}{y}={0}$

$\displaystyle{2}{\left({0}\right)}+{3}{y}={0}$

$\displaystyle{3}{y}={0}$

$\displaystyle{y}={0}$

Plug in the value of x and y in equation 2,

$\displaystyle{4}{\left({0}\right)}+{3}{\left({0}\right)}-{z}={0}$

$\displaystyle-{z}={0}$

$\displaystyle{z}={0}$

Solutions of the equations are x=0, y=0 and z=0

Eq1: $\displaystyle{2}{x}+{3}{y}={0}$

Eq2: $\displaystyle{4}{x}+{3}{y}-{z}={0}$

Eq3: $\displaystyle{8}{x}+{3}{y}+{3}{z}={0}$

Multiply equation 2 by 3,

$\displaystyle{12}{x}+{9}{y}-{3}{z}={0}$

Now add the above equation and equation 3,

$\displaystyle{\left({8}{x}+{3}{y}+{3}{z}\right)}+{\left({12}{x}+{9}{y}-{3}{z}\right)}={0}$

$\displaystyle{8}{x}+{12}{x}+{3}{y}+{9}{y}+{3}{z}-{3}{z}={0}$

$\displaystyle{20}{x}+{12}{y}={0}$

$\displaystyle{4}{\left({5}{x}+{3}{y}\right)}={0}$

$\displaystyle{5}{x}+{3}{y}={0}$

Now let's solve the above equation and equation 1,

$\displaystyle{2}{x}+{3}{y}={0}$

$\displaystyle{5}{x}+{3}{y}={0}$

Subtract the above equations

$\displaystyle{2}{x}-{5}{x}={0}$

$\displaystyle-{3}{x}={0}$

$\displaystyle{x}={0}$

Substitute back the value of x,

$\displaystyle{2}{x}+{3}{y}={0}$

$\displaystyle{2}{\left({0}\right)}+{3}{y}={0}$

$\displaystyle{3}{y}={0}$

$\displaystyle{y}={0}$

Plug in the value of x and y in equation 2,

$\displaystyle{4}{\left({0}\right)}+{3}{\left({0}\right)}-{z}={0}$

$\displaystyle-{z}={0}$

$\displaystyle{z}={0}$

Solutions of the equations are x=0, y=0 and z=0

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)