Solve {l}{4x+7y+8z=143}{6x+y+z=52}{3x+5y+4z=91}

$$\left\{ \begin{array}{l}{ 4 x + 7 y + 8 z = 143 }\\{ 6 x + y + z = 52 }\\{ 3 x + 5 y + 4 z = 91 }\end{array} \right.$$

Solve for x, y, z

$$x=6$$

$$y=9$$

$$z=7$$

Short Steps Using Substitution

Reorder the equations.

$$6x+y+z=52 4x+7y+8z=143 3x+5y+4z=91$$

Solve 6x+y+z=52 for y.

$$y=-6x-z+52$$

Substitute -6x-z+52 for y in the second and third equation.

$$4x+7\left(-6x-z+52\right)+8z=143 3x+5\left(-6x-z+52\right)+4z=91$$

Solve these equations for x and z respectively.

$$x=\frac{221}{38}+\frac{1}{38}z z=-27x+169$$

Substitute \frac{221}{38}+\frac{1}{38}z for x in the equation z=-27x+169.

$$z=-27\left(\frac{221}{38}+\frac{1}{38}z\right)+169$$

Solve z=-27\left(\frac{221}{38}+\frac{1}{38}z\right)+169 for z.

$$z=7$$

Substitute 7 for z in the equation x=\frac{221}{38}+\frac{1}{38}z.

$$x=\frac{221}{38}+\frac{1}{38}\times 7$$

Calculate x from x=\frac{221}{38}+\frac{1}{38}\times 7.

$$x=6$$

Substitute 6 for x and 7 for z in the equation y=-6x-z+52.

$$y=-6\times 6-7+52$$

Calculate y from y=-6\times 6-7+52.

$$y=9$$

The system is now solved.

$$x=6 y=9 z=7$$