Resolver {l}{5x+2y+3z=8}{7x+8y+2z=1}{9x+4y+9z=7}

$$\left. \begin{array} { l } { 5 x + 2 y + 3 z = 8 } \\ { 7 x + 8 y + 2 z = 1 } \\ { 9 x + 4 y + 9 z = 7 } \end{array} \right.$$

Resolver x, y, z

$$x = \frac{183}{49} = 3\frac{36}{49} \approx 3.734693878$$

$$y = -\frac{265}{98} = -2\frac{69}{98} \approx -2.704081633$$

$$z = -\frac{86}{49} = -1\frac{37}{49} \approx -1.755102041$$

Pasos breves que usan substitución

Despexa x en 5x+2y+3z=8.

$$x=-\frac{2}{5}y-\frac{3}{5}z+\frac{8}{5}$$

Substitúe -\frac{2}{5}y-\frac{3}{5}z+\frac{8}{5} por x na segunda e na terceira ecuación.

$$7\left(-\frac{2}{5}y-\frac{3}{5}z+\frac{8}{5}\right)+8y+2z=1 9\left(-\frac{2}{5}y-\frac{3}{5}z+\frac{8}{5}\right)+4y+9z=7$$

Despexa y e z respectivamente nestas ecuacións.

$$y=-\frac{51}{26}+\frac{11}{26}z z=-\frac{37}{18}-\frac{1}{9}y$$

Substitúe y por -\frac{51}{26}+\frac{11}{26}z na ecuación z=-\frac{37}{18}-\frac{1}{9}y.

$$z=-\frac{37}{18}-\frac{1}{9}\left(-\frac{51}{26}+\frac{11}{26}z\right)$$

Despexa z en z=-\frac{37}{18}-\frac{1}{9}\left(-\frac{51}{26}+\frac{11}{26}z\right).

$$z=-\frac{86}{49}$$

Substitúe z por -\frac{86}{49} na ecuación y=-\frac{51}{26}+\frac{11}{26}z.

$$y=-\frac{51}{26}+\frac{11}{26}\left(-\frac{86}{49}\right)$$

Calcular y tendo en conta que y=-\frac{51}{26}+\frac{11}{26}\left(-\frac{86}{49}\right).

$$y=-\frac{265}{98}$$

Substitúe -\frac{265}{98} por y e -\frac{86}{49} por z na ecuación x=-\frac{2}{5}y-\frac{3}{5}z+\frac{8}{5}.

$$x=-\frac{2}{5}\left(-\frac{265}{98}\right)-\frac{3}{5}\left(-\frac{86}{49}\right)+\frac{8}{5}$$

Calcular x tendo en conta que x=-\frac{2}{5}\left(-\frac{265}{98}\right)-\frac{3}{5}\left(-\frac{86}{49}\right)+\frac{8}{5}.

$$x=\frac{183}{49}$$

O sistema xa funciona correctamente.

$$x=\frac{183}{49} y=-\frac{265}{98} z=-\frac{86}{49}$$