Solve {c}{x-3y+4z-5=0}{2x+y+z-3=0}{4x+3y+5z-1=0}

$$\left\{ \begin{array} { c } { x - 3 y + 4 z - 5 = 0 } \\ { 2 x + y + z - 3 = 0 } \\ { 4 x + 3 y + 5 z - 1 = 0 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-2$$

$$z=-1$$

Short Steps Using Substitution

Solve x-3y+4z-5=0 for x.

$$x=3y-4z+5$$

Substitute 3y-4z+5 for x in the second and third equation.

$$2\left(3y-4z+5\right)+y+z-3=0 4\left(3y-4z+5\right)+3y+5z-1=0$$

Solve these equations for y and z respectively.

$$y=-1+z z=\frac{15}{11}y+\frac{19}{11}$$

Substitute -1+z for y in the equation z=\frac{15}{11}y+\frac{19}{11}.

$$z=\frac{15}{11}\left(-1+z\right)+\frac{19}{11}$$

Solve z=\frac{15}{11}\left(-1+z\right)+\frac{19}{11} for z.

$$z=-1$$

Substitute -1 for z in the equation y=-1+z.

$$y=-1-1$$

Calculate y from y=-1-1.

$$y=-2$$

Substitute -2 for y and -1 for z in the equation x=3y-4z+5.

$$x=3\left(-2\right)-4\left(-1\right)+5$$

Calculate x from x=3\left(-2\right)-4\left(-1\right)+5.

$$x=3$$

The system is now solved.

$$x=3 y=-2 z=-1$$