Solve {l}{11x+3y+5z=27}{22x+8y+6z=38}{37x+9y+12z=52}

$$\left. \begin{array} { l } { 11 x + 3 y + 5 z = 27 } \\ { 22 x + 8 y + 6 z = 38 } \\ { 37 x + 9 y + 12 z = 52 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{166}{77} = -2\frac{12}{77} \approx -2.155844156$$

$$y = \frac{430}{77} = 5\frac{45}{77} \approx 5.584415584$$

$$z = \frac{523}{77} = 6\frac{61}{77} \approx 6.792207792$$

Short Steps Using Substitution

Solve 11x+3y+5z=27 for x.

$$x=-\frac{3}{11}y-\frac{5}{11}z+\frac{27}{11}$$

Substitute -\frac{3}{11}y-\frac{5}{11}z+\frac{27}{11} for x in the second and third equation.

$$22\left(-\frac{3}{11}y-\frac{5}{11}z+\frac{27}{11}\right)+8y+6z=38 37\left(-\frac{3}{11}y-\frac{5}{11}z+\frac{27}{11}\right)+9y+12z=52$$

Solve these equations for y and z respectively.

$$y=-8+2z z=\frac{427}{53}-\frac{12}{53}y$$

Substitute -8+2z for y in the equation z=\frac{427}{53}-\frac{12}{53}y.

$$z=\frac{427}{53}-\frac{12}{53}\left(-8+2z\right)$$

Solve z=\frac{427}{53}-\frac{12}{53}\left(-8+2z\right) for z.

$$z=\frac{523}{77}$$

Substitute \frac{523}{77} for z in the equation y=-8+2z.

$$y=-8+2\times \frac{523}{77}$$

Calculate y from y=-8+2\times \frac{523}{77}.

$$y=\frac{430}{77}$$

Substitute \frac{430}{77} for y and \frac{523}{77} for z in the equation x=-\frac{3}{11}y-\frac{5}{11}z+\frac{27}{11}.

$$x=-\frac{3}{11}\times \frac{430}{77}-\frac{5}{11}\times \frac{523}{77}+\frac{27}{11}$$

Calculate x from x=-\frac{3}{11}\times \frac{430}{77}-\frac{5}{11}\times \frac{523}{77}+\frac{27}{11}.

$$x=-\frac{166}{77}$$

The system is now solved.

$$x=-\frac{166}{77} y=\frac{430}{77} z=\frac{523}{77}$$