Solve {c}{4x+3y+2z=25}{3x-2y+5z=20}{10x-5y+3z=17}

$$\left. \begin{array} { c } { 4 x + 3 y + 2 z = 25 } \\ { 3 x - 2 y + 5 z = 20 } \\ { 10 x - 5 y + 3 z = 17 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=3$$

$$z=4$$

Short Steps Using Substitution

Solve 4x+3y+2z=25 for x.

$$x=-\frac{3}{4}y-\frac{1}{2}z+\frac{25}{4}$$

Substitute -\frac{3}{4}y-\frac{1}{2}z+\frac{25}{4} for x in the second and third equation.

$$3\left(-\frac{3}{4}y-\frac{1}{2}z+\frac{25}{4}\right)-2y+5z=20 10\left(-\frac{3}{4}y-\frac{1}{2}z+\frac{25}{4}\right)-5y+3z=17$$

Solve these equations for y and z respectively.

$$y=-\frac{5}{17}+\frac{14}{17}z z=\frac{91}{4}-\frac{25}{4}y$$

Substitute -\frac{5}{17}+\frac{14}{17}z for y in the equation z=\frac{91}{4}-\frac{25}{4}y.

$$z=\frac{91}{4}-\frac{25}{4}\left(-\frac{5}{17}+\frac{14}{17}z\right)$$

Solve z=\frac{91}{4}-\frac{25}{4}\left(-\frac{5}{17}+\frac{14}{17}z\right) for z.

$$z=4$$

Substitute 4 for z in the equation y=-\frac{5}{17}+\frac{14}{17}z.

$$y=-\frac{5}{17}+\frac{14}{17}\times 4$$

Calculate y from y=-\frac{5}{17}+\frac{14}{17}\times 4.

$$y=3$$

Substitute 3 for y and 4 for z in the equation x=-\frac{3}{4}y-\frac{1}{2}z+\frac{25}{4}.

$$x=-\frac{3}{4}\times 3-\frac{1}{2}\times 4+\frac{25}{4}$$

Calculate x from x=-\frac{3}{4}\times 3-\frac{1}{2}\times 4+\frac{25}{4}.

$$x=2$$

The system is now solved.

$$x=2 y=3 z=4$$