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

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

Solve for x, y, z

$$x=-1$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$z = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for z and x respectively.

$$z=\frac{1}{2}-2x x=-1$$

Substitute -1 for x in the equation z=\frac{1}{2}-2x.

$$z=\frac{1}{2}-2\left(-1\right)$$

Calculate z from z=\frac{1}{2}-2\left(-1\right).

$$z=\frac{5}{2}$$

Substitute \frac{5}{2} for z and -1 for x in the equation y=-3x-z+1.

$$y=-3\left(-1\right)-\frac{5}{2}+1$$

Calculate y from y=-3\left(-1\right)-\frac{5}{2}+1.

$$y=\frac{3}{2}$$

The system is now solved.

$$x=-1 y=\frac{3}{2} z=\frac{5}{2}$$