Solve {l}{2x+3y+4z=3}{7x-5y=24}{4x+11z=39}

$$\left\{ \begin{array} { l } { 2 x + 3 y + 4 z = 3 } \\ { 7 x - 5 y = 24 } \\ { 4 x + 11 z = 39 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{59}{87}\approx 0.67816092$$

$$y = -\frac{335}{87} = -3\frac{74}{87} \approx -3.850574713$$

$$z = \frac{287}{87} = 3\frac{26}{87} \approx 3.298850575$$

Short Steps Using Substitution

Solve 2x+3y+4z=3 for x.

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

Substitute -\frac{3}{2}y-2z+\frac{3}{2} for x in the second and third equation.

$$7\left(-\frac{3}{2}y-2z+\frac{3}{2}\right)-5y=24 4\left(-\frac{3}{2}y-2z+\frac{3}{2}\right)+11z=39$$

Solve these equations for y and z respectively.

$$y=-\frac{27}{31}-\frac{28}{31}z z=11+2y$$

Substitute -\frac{27}{31}-\frac{28}{31}z for y in the equation z=11+2y.

$$z=11+2\left(-\frac{27}{31}-\frac{28}{31}z\right)$$

Solve z=11+2\left(-\frac{27}{31}-\frac{28}{31}z\right) for z.

$$z=\frac{287}{87}$$

Substitute \frac{287}{87} for z in the equation y=-\frac{27}{31}-\frac{28}{31}z.

$$y=-\frac{27}{31}-\frac{28}{31}\times \frac{287}{87}$$

Calculate y from y=-\frac{27}{31}-\frac{28}{31}\times \frac{287}{87}.

$$y=-\frac{335}{87}$$

Substitute -\frac{335}{87} for y and \frac{287}{87} for z in the equation x=-\frac{3}{2}y-2z+\frac{3}{2}.

$$x=-\frac{3}{2}\left(-\frac{335}{87}\right)-2\times \frac{287}{87}+\frac{3}{2}$$

Calculate x from x=-\frac{3}{2}\left(-\frac{335}{87}\right)-2\times \frac{287}{87}+\frac{3}{2}.

$$x=\frac{59}{87}$$

The system is now solved.

$$x=\frac{59}{87} y=-\frac{335}{87} z=\frac{287}{87}$$