Solve {l}{x+y+z=525}{23x+30y+40z=1250}{x+2y+4z=596}

$$\left\{ \begin{array} { l } { x + y + z = 525 } \\ { 23 x + 30 y + 40 z = 1250 } \\ { x + 2 y + 4 z = 596 } \end{array} \right.$$

Solve for x, y, z

$$x=6115$$

$$y = -\frac{16841}{2} = -8420\frac{1}{2} = -8420.5$$

$$z = \frac{5661}{2} = 2830\frac{1}{2} = 2830.5$$

Short Steps Using Substitution

Solve x+y+z=525 for x.

$$x=-y-z+525$$

Substitute -y-z+525 for x in the second and third equation.

$$23\left(-y-z+525\right)+30y+40z=1250 -y-z+525+2y+4z=596$$

Solve these equations for y and z respectively.

$$y=-\frac{17}{7}z-\frac{10825}{7} z=-\frac{1}{3}y+\frac{71}{3}$$

Substitute -\frac{17}{7}z-\frac{10825}{7} for y in the equation z=-\frac{1}{3}y+\frac{71}{3}.

$$z=-\frac{1}{3}\left(-\frac{17}{7}z-\frac{10825}{7}\right)+\frac{71}{3}$$

Solve z=-\frac{1}{3}\left(-\frac{17}{7}z-\frac{10825}{7}\right)+\frac{71}{3} for z.

$$z=\frac{5661}{2}$$

Substitute \frac{5661}{2} for z in the equation y=-\frac{17}{7}z-\frac{10825}{7}.

$$y=-\frac{17}{7}\times \frac{5661}{2}-\frac{10825}{7}$$

Calculate y from y=-\frac{17}{7}\times \frac{5661}{2}-\frac{10825}{7}.

$$y=-\frac{16841}{2}$$

Substitute -\frac{16841}{2} for y and \frac{5661}{2} for z in the equation x=-y-z+525.

$$x=-\left(-\frac{16841}{2}\right)-\frac{5661}{2}+525$$

Calculate x from x=-\left(-\frac{16841}{2}\right)-\frac{5661}{2}+525.

$$x=6115$$

The system is now solved.

$$x=6115 y=-\frac{16841}{2} z=\frac{5661}{2}$$