Solve {c}{0.149=0.19x+0.405y+z}{0.036=0.045x+0.11y+z}{1.0=x+y+z}

$$\left. \begin{array} { c } { 0.149 = 0.19 x + 0.405 y + z } \\ { 0.036 = 0.045 x + 0.11 y + z } \\ { 1.0 = x + y + z } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{36762}{30535} = 1\frac{6227}{30535} \approx 1.203929916$$

$$y=-\frac{6373}{30535}\approx -0.208711315$$

$$z=\frac{146}{30535}\approx 0.004781398$$

Short Steps Using Substitution

Solve 0.149=0.19x+0.405y+z for z.

$$z=-0.19x+0.149-0.405y$$

Substitute -0.19x+0.149-0.405y for z in the second and third equation.

$$0.036=0.045x+0.11y-0.19x+0.149-0.405y 1=x+y-0.19x+0.149-0.405y$$

Solve these equations for y and x respectively.

$$y=\frac{113}{295}-\frac{29}{59}x x=\frac{851}{810}-\frac{119}{162}y$$

Substitute \frac{113}{295}-\frac{29}{59}x for y in the equation x=\frac{851}{810}-\frac{119}{162}y.

$$x=\frac{851}{810}-\frac{119}{162}\left(\frac{113}{295}-\frac{29}{59}x\right)$$

Solve x=\frac{851}{810}-\frac{119}{162}\left(\frac{113}{295}-\frac{29}{59}x\right) for x.

$$x=\frac{36762}{30535}$$

Substitute \frac{36762}{30535} for x in the equation y=\frac{113}{295}-\frac{29}{59}x.

$$y=\frac{113}{295}-\frac{29}{59}\times \frac{36762}{30535}$$

Calculate y from y=\frac{113}{295}-\frac{29}{59}\times \frac{36762}{30535}.

$$y=-\frac{6373}{30535}$$

Substitute -\frac{6373}{30535} for y and \frac{36762}{30535} for x in the equation z=-0.19x+0.149-0.405y.

$$z=-0.19\times \frac{36762}{30535}+0.149-0.405\left(-\frac{6373}{30535}\right)$$

Calculate z from z=-0.19\times \frac{36762}{30535}+0.149-0.405\left(-\frac{6373}{30535}\right).

$$z=\frac{146}{30535}$$

The system is now solved.

$$x=\frac{36762}{30535} y=-\frac{6373}{30535} z=\frac{146}{30535}$$