Solve {l}{5x+4y+3z=20}{4x+6y+2z=7}{3x-y+5z=18}

$$\left. \begin{array} { l } { 5 x + 4 y + 3 z = 20 } \\ { 4 x + 6 y + 2 z = 7 } \\ { 3 x - y + 5 z = 18 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{299}{38} = 7\frac{33}{38} \approx 7.868421053$$

$$y = -\frac{66}{19} = -3\frac{9}{19} \approx -3.473684211$$

$$z = -\frac{69}{38} = -1\frac{31}{38} \approx -1.815789474$$

Short Steps Using Substitution

Reorder the equations.

$$3x-y+5z=18 4x+6y+2z=7 5x+4y+3z=20$$

Solve 3x-y+5z=18 for y.

$$y=3x+5z-18$$

Substitute 3x+5z-18 for y in the second and third equation.

$$4x+6\left(3x+5z-18\right)+2z=7 5x+4\left(3x+5z-18\right)+3z=20$$

Solve these equations for x and z respectively.

$$x=-\frac{16}{11}z+\frac{115}{22} z=-\frac{17}{23}x+4$$

Substitute -\frac{16}{11}z+\frac{115}{22} for x in the equation z=-\frac{17}{23}x+4.

$$z=-\frac{17}{23}\left(-\frac{16}{11}z+\frac{115}{22}\right)+4$$

Solve z=-\frac{17}{23}\left(-\frac{16}{11}z+\frac{115}{22}\right)+4 for z.

$$z=-\frac{69}{38}$$

Substitute -\frac{69}{38} for z in the equation x=-\frac{16}{11}z+\frac{115}{22}.

$$x=-\frac{16}{11}\left(-\frac{69}{38}\right)+\frac{115}{22}$$

Calculate x from x=-\frac{16}{11}\left(-\frac{69}{38}\right)+\frac{115}{22}.

$$x=\frac{299}{38}$$

Substitute \frac{299}{38} for x and -\frac{69}{38} for z in the equation y=3x+5z-18.

$$y=3\times \frac{299}{38}+5\left(-\frac{69}{38}\right)-18$$

Calculate y from y=3\times \frac{299}{38}+5\left(-\frac{69}{38}\right)-18.

$$y=-\frac{66}{19}$$

The system is now solved.

$$x=\frac{299}{38} y=-\frac{66}{19} z=-\frac{69}{38}$$