Solve {l}{9x+2y+4z=21,210}{4x+4y+4z=19,027}{y=2,500}

$$\left. \begin{array} { l } { 9 x + 2 y + 4 z = 21,210 } \\ { 4 x + 4 y + 4 z = 19,027 } \\ { y = 2,500 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{7183}{5} = 1436\frac{3}{5} = 1436.6$$

$$y=2500$$

$$z = \frac{16403}{20} = 820\frac{3}{20} = 820.15$$

Short Steps Using Substitution

Reorder the equations.

$$y=2500 4x+4y+4z=19027 9x+2y+4z=21210$$

Substitute 2500 for y in the second and third equation.

$$4x+4\times 2500+4z=19027 9x+2\times 2500+4z=21210$$

Solve these equations for x and z respectively.

$$x=\frac{9027}{4}-z z=-\frac{9}{4}x+\frac{8105}{2}$$

Substitute \frac{9027}{4}-z for x in the equation z=-\frac{9}{4}x+\frac{8105}{2}.

$$z=-\frac{9}{4}\left(\frac{9027}{4}-z\right)+\frac{8105}{2}$$

Solve z=-\frac{9}{4}\left(\frac{9027}{4}-z\right)+\frac{8105}{2} for z.

$$z=\frac{16403}{20}$$

Substitute \frac{16403}{20} for z in the equation x=\frac{9027}{4}-z.

$$x=\frac{9027}{4}-\frac{16403}{20}$$

Calculate x from x=\frac{9027}{4}-\frac{16403}{20}.

$$x=\frac{7183}{5}$$

The system is now solved.

$$x=\frac{7183}{5} y=2500 z=\frac{16403}{20}$$