Solve {l}{7x+3y-z=7}{16x-5y+2z=12}{6x+15y-7z=14}

$$\left. \begin{array} { l } { 7 x + 3 y - z = 7 } \\ { 16 x - 5 y + 2 z = 12 } \\ { 6 x + 15 y - 7 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{121}{137}\approx 0.883211679$$

$$y=-\frac{68}{137}\approx -0.496350365$$

$$z = -\frac{316}{137} = -2\frac{42}{137} \approx -2.306569343$$

Short Steps Using Substitution

Solve 7x+3y-z=7 for z.

$$z=7x+3y-7$$

Substitute 7x+3y-7 for z in the second and third equation.

$$16x-5y+2\left(7x+3y-7\right)=12 6x+15y-7\left(7x+3y-7\right)=14$$

Solve these equations for y and x respectively.

$$y=26-30x x=-\frac{6}{43}y+\frac{35}{43}$$

Substitute 26-30x for y in the equation x=-\frac{6}{43}y+\frac{35}{43}.

$$x=-\frac{6}{43}\left(26-30x\right)+\frac{35}{43}$$

Solve x=-\frac{6}{43}\left(26-30x\right)+\frac{35}{43} for x.

$$x=\frac{121}{137}$$

Substitute \frac{121}{137} for x in the equation y=26-30x.

$$y=26-30\times \frac{121}{137}$$

Calculate y from y=26-30\times \frac{121}{137}.

$$y=-\frac{68}{137}$$

Substitute -\frac{68}{137} for y and \frac{121}{137} for x in the equation z=7x+3y-7.

$$z=7\times \frac{121}{137}+3\left(-\frac{68}{137}\right)-7$$

Calculate z from z=7\times \frac{121}{137}+3\left(-\frac{68}{137}\right)-7.

$$z=-\frac{316}{137}$$

The system is now solved.

$$x=\frac{121}{137} y=-\frac{68}{137} z=-\frac{316}{137}$$