Solve {c}{3x+2y+20=6}{4x-y+2z=5}{7x+3y-3z=7}

$$\left. \begin{array} { c } { 3 x + 2 y + 20 = 6 } \\ { 4 x - y + 2 z = 5 } \\ { 7 x + 3 y - 3 z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{100}{43} = 2\frac{14}{43} \approx 2.325581395$$

$$y = -\frac{451}{43} = -10\frac{21}{43} \approx -10.488372093$$

$$z = -\frac{318}{43} = -7\frac{17}{43} \approx -7.395348837$$

Short Steps Using Substitution

Reorder the equations.

$$4x-y+2z=5 3x+2y+20=6 7x+3y-3z=7$$

Solve 4x-y+2z=5 for y.

$$y=4x+2z-5$$

Substitute 4x+2z-5 for y in the second and third equation.

$$3x+2\left(4x+2z-5\right)+20=6 7x+3\left(4x+2z-5\right)-3z=7$$

Solve these equations for x and z respectively.

$$x=-\frac{4}{11}-\frac{4}{11}z z=-\frac{19}{3}x+\frac{22}{3}$$

Substitute -\frac{4}{11}-\frac{4}{11}z for x in the equation z=-\frac{19}{3}x+\frac{22}{3}.

$$z=-\frac{19}{3}\left(-\frac{4}{11}-\frac{4}{11}z\right)+\frac{22}{3}$$

Solve z=-\frac{19}{3}\left(-\frac{4}{11}-\frac{4}{11}z\right)+\frac{22}{3} for z.

$$z=-\frac{318}{43}$$

Substitute -\frac{318}{43} for z in the equation x=-\frac{4}{11}-\frac{4}{11}z.

$$x=-\frac{4}{11}-\frac{4}{11}\left(-\frac{318}{43}\right)$$

Calculate x from x=-\frac{4}{11}-\frac{4}{11}\left(-\frac{318}{43}\right).

$$x=\frac{100}{43}$$

Substitute \frac{100}{43} for x and -\frac{318}{43} for z in the equation y=4x+2z-5.

$$y=4\times \frac{100}{43}+2\left(-\frac{318}{43}\right)-5$$

Calculate y from y=4\times \frac{100}{43}+2\left(-\frac{318}{43}\right)-5.

$$y=-\frac{451}{43}$$

The system is now solved.

$$x=\frac{100}{43} y=-\frac{451}{43} z=-\frac{318}{43}$$