Solve {l}{3x+4y+z=7}{4x+y+z=10}{x+y+z=4}

$$\left. \begin{array} { l } { 3 x + 4 y + z = 7 } \\ { 4 x + y + z = 10 } \\ { x + y + z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-\frac{1}{3}\approx -0.333333333$$

$$z = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Short Steps Using Substitution

Solve 3x+4y+z=7 for z.

$$z=-3x-4y+7$$

Substitute -3x-4y+7 for z in the second and third equation.

$$4x+y-3x-4y+7=10 x+y-3x-4y+7=4$$

Solve these equations for y and x respectively.

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

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

$$x=\frac{3}{2}-\frac{3}{2}\left(\frac{1}{3}x-1\right)$$

Solve x=\frac{3}{2}-\frac{3}{2}\left(\frac{1}{3}x-1\right) for x.

$$x=2$$

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

$$y=\frac{1}{3}\times 2-1$$

Calculate y from y=\frac{1}{3}\times 2-1.

$$y=-\frac{1}{3}$$

Substitute -\frac{1}{3} for y and 2 for x in the equation z=-3x-4y+7.

$$z=-3\times 2-4\left(-\frac{1}{3}\right)+7$$

Calculate z from z=-3\times 2-4\left(-\frac{1}{3}\right)+7.

$$z=\frac{7}{3}$$

The system is now solved.

$$x=2 y=-\frac{1}{3} z=\frac{7}{3}$$