Solve {l}{0.70740x+0.15290y+0.13980z=1399}{0.08840x+0.73800y+0.17360z=991}{0.00223x+0.00571y+0.03881z=6.5}

$$\left. \begin{array} { l } { 0.70740 x + 0.15290 y + 0.13980 z = 1399 } \\ { 0.08840 x + 0.73800 y + 0.17360 z = 991 } \\ { 0.00223 x + 0.00571 y + 0.03881 z = 6.5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{82738564887500}{47337735431} = 1747\frac{39541089543}{47337735431} \approx 1747.835297447$$

$$y = \frac{54805385590000}{47337735431} = 1157\frac{35625696333}{47337735431} \approx 1157.752585564$$

$$z = -\frac{4889215952500}{47337735431} = -103\frac{13429203107}{47337735431} \approx -103.283689175$$

Short Steps Using Substitution

Solve 0.7074x+0.1529y+0.1398z=1399 for x.

$$x=-\frac{1529}{7074}y-\frac{233}{1179}z+\frac{6995000}{3537}$$

Substitute -\frac{1529}{7074}y-\frac{233}{1179}z+\frac{6995000}{3537} for x in the second and third equation.

$$0.0884\left(-\frac{1529}{7074}y-\frac{233}{1179}z+\frac{6995000}{3537}\right)+0.738y+0.1736z=991 0.00223\left(-\frac{1529}{7074}y-\frac{233}{1179}z+\frac{6995000}{3537}\right)+0.00571y+0.03881z=6.5$$

Solve these equations for y and z respectively.

$$y=-\frac{2761158}{12713621}z+\frac{14434045000}{12713621} z=-\frac{3698287}{27142440}y+\frac{36958250}{678561}$$

Substitute -\frac{2761158}{12713621}z+\frac{14434045000}{12713621} for y in the equation z=-\frac{3698287}{27142440}y+\frac{36958250}{678561}.

$$z=-\frac{3698287}{27142440}\left(-\frac{2761158}{12713621}z+\frac{14434045000}{12713621}\right)+\frac{36958250}{678561}$$

Solve z=-\frac{3698287}{27142440}\left(-\frac{2761158}{12713621}z+\frac{14434045000}{12713621}\right)+\frac{36958250}{678561} for z.

$$z=-\frac{4889215952500}{47337735431}$$

Substitute -\frac{4889215952500}{47337735431} for z in the equation y=-\frac{2761158}{12713621}z+\frac{14434045000}{12713621}.

$$y=-\frac{2761158}{12713621}\left(-\frac{4889215952500}{47337735431}\right)+\frac{14434045000}{12713621}$$

Calculate y from y=-\frac{2761158}{12713621}\left(-\frac{4889215952500}{47337735431}\right)+\frac{14434045000}{12713621}.

$$y=\frac{54805385590000}{47337735431}$$

Substitute \frac{54805385590000}{47337735431} for y and -\frac{4889215952500}{47337735431} for z in the equation x=-\frac{1529}{7074}y-\frac{233}{1179}z+\frac{6995000}{3537}.

$$x=-\frac{1529}{7074}\times \frac{54805385590000}{47337735431}-\frac{233}{1179}\left(-\frac{4889215952500}{47337735431}\right)+\frac{6995000}{3537}$$

Calculate x from x=-\frac{1529}{7074}\times \frac{54805385590000}{47337735431}-\frac{233}{1179}\left(-\frac{4889215952500}{47337735431}\right)+\frac{6995000}{3537}.

$$x=\frac{82738564887500}{47337735431}$$

The system is now solved.

$$x=\frac{82738564887500}{47337735431} y=\frac{54805385590000}{47337735431} z=-\frac{4889215952500}{47337735431}$$