Solve {l}{x+3y+z=1}{3x+3y+2z=4}{4x+7y+3z=5}

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

Solve for x, y, z

$$x=2$$

$$y=0$$

$$z=-1$$

Short Steps Using Substitution

Solve x+3y+z=1 for x.

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

Substitute -3y-z+1 for x in the second and third equation.

$$3\left(-3y-z+1\right)+3y+2z=4 4\left(-3y-z+1\right)+7y+3z=5$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{6}z-\frac{1}{6} z=-5y-1$$

Substitute -\frac{1}{6}z-\frac{1}{6} for y in the equation z=-5y-1.

$$z=-5\left(-\frac{1}{6}z-\frac{1}{6}\right)-1$$

Solve z=-5\left(-\frac{1}{6}z-\frac{1}{6}\right)-1 for z.

$$z=-1$$

Substitute -1 for z in the equation y=-\frac{1}{6}z-\frac{1}{6}.

$$y=-\frac{1}{6}\left(-1\right)-\frac{1}{6}$$

Calculate y from y=-\frac{1}{6}\left(-1\right)-\frac{1}{6}.

$$y=0$$

Substitute 0 for y and -1 for z in the equation x=-3y-z+1.

$$x=-3\times 0-\left(-1\right)+1$$

Calculate x from x=-3\times 0-\left(-1\right)+1.

$$x=2$$

The system is now solved.

$$x=2 y=0 z=-1$$