Find the value of $$ z $$ in the solution of the following system: $$ \left\{\begin{array}{l}4 x+1 y+3 z=37 \ 3 x-3 y-2 z=-35 \ 1 x+5 y+5 z=91\end{array} ight. $$

Question

Find the value of $$ z $$ in the solution of the following system: $$ \left\{\begin{array}{l}4 x+1 y+3 z=37 \ 3 x-3 y-2 z=-35 \ 1 x+5 y+5 z=91\end{array}ight. $$

UpStudy AI · Accepted Answer

Solve the system of equations $$ 4x+1y+3z=37;3x-3y-2z=-35;1x+5y+5z=91 $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}4x+1	imes y+3z=37\3x-3y-2z=-35\1	imes x+5y+5z=91\end{array}ight.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}4x+y+3z=37\3x-3y-2z=-35\x+5y+5z=91\end{array}ight.$$
- step2: Solve the equation:
  $$\left\{ \begin{array}{l}y=37-4x-3z\3x-3y-2z=-35\x+5y+5z=91\end{array}ight.$$
- step3: Substitute the value of $$y:$$
  $$\left\{ \begin{array}{l}3x-3\left(37-4x-3zight)-2z=-35\x+5\left(37-4x-3zight)+5z=91\end{array}ight.$$
- step4: Simplify:
  $$\left\{ \begin{array}{l}15x-111+7z=-35\-19x+185-10z=91\end{array}ight.$$
- step5: Solve the equation:
  $$\left\{ \begin{array}{l}x=\frac{76-7z}{15}\-19x+185-10z=91\end{array}ight.$$
- step6: Substitute the value of $$x:$$
  $$-19	imes \frac{76-7z}{15}+185-10z=91$$
- step7: Simplify:
  $$-\frac{19\left(76-7zight)}{15}+185-10z=91$$
- step8: Multiply both sides of the equation by LCD:
  $$\left(-\frac{19\left(76-7zight)}{15}+185-10zight)	imes 15=91	imes 15$$
- step9: Simplify the equation:
  $$1331-17z=1365$$
- step10: Move the constant to the right side:
  $$-17z=1365-1331$$
- step11: Subtract the numbers:
  $$-17z=34$$
- step12: Change the signs:
  $$17z=-34$$
- step13: Divide both sides:
  $$\frac{17z}{17}=\frac{-34}{17}$$
- step14: Divide the numbers:
  $$z=-2$$
- step15: Substitute the value of $$z:$$
  $$x=\frac{76-7\left(-2ight)}{15}$$
- step16: Calculate:
  $$x=6$$
- step17: Substitute the values of $$x,z:$$
  $$y=37-4	imes 6-3\left(-2ight)$$
- step18: Calculate:
  $$y=19$$
- step19: Calculate:
  $$\left\{ \begin{array}{l}x=6\y=19\z=-2\end{array}ight.$$
- step20: Check the solution:
  $$\left\{ \begin{array}{l}x=6\y=19\z=-2\end{array}ight.$$
- step21: Rewrite:
  $$\left(x,y,zight) = \left(6,19,-2ight)$$
The solution to the system of equations is $$ (x, y, z) = (6, 19, -2) $$. Therefore, the value of $$ z $$ in the solution is $$ -2 $$.
$$ z = -2 $$

Find the value of in the solution of the following system:

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