An investment of $$ \$ 108,000 $$ was made by a business club. The investment was split into three parts and lasted for one year. The first part of the investment earned $$ 8 \% $$ interest, the second $$ 6 \% $$, and the third $$ 9 \% $$. Total interest from the investments was $$ \$ 8520 $$. The interest from the first investment was 6 times the interest from the second. Find the amounts of the three parts of the investment.

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Let's denote the amounts of the three parts of the investment as $$ x $$, $$ y $$, and $$ z $$ respectively.

Given:
1. The total investment is $$ \$108,000 $$, so $$ x + y + z = 108,000 $$.
2. The interest from the first investment is 6 times the interest from the second, so $$ 0.08x = 6(0.06y) $$.
3. The total interest from the investments is $$ \$8520 $$, so $$ 0.08x + 0.06y + 0.09z = 8520 $$.

We can solve this system of equations to find the amounts of the three parts of the investment.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}x+y+z=108000\0.08x=6	imes 0.06y\0.08x+0.06y+0.09z=8520\end{array}ight.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}x+y+z=108000\0.08x=0.36y\0.08x+0.06y+0.09z=8520\end{array}ight.$$
- step2: Solve the equation:
  $$\left\{ \begin{array}{l}x=108000-y-z\0.08x=0.36y\0.08x+0.06y+0.09z=8520\end{array}ight.$$
- step3: Substitute the value of $$x:$$
  $$\left\{ \begin{array}{l}0.08\left(108000-y-zight)=0.36y\0.08\left(108000-y-zight)+0.06y+0.09z=8520\end{array}ight.$$
- step4: Simplify:
  $$\left\{ \begin{array}{l}0.08\left(108000-y-zight)=0.36y\8640-0.02y+0.01z=8520\end{array}ight.$$
- step5: Solve the equation:
  $$\left\{ \begin{array}{l}y=\frac{216000-2z}{11}\8640-0.02y+0.01z=8520\end{array}ight.$$
- step6: Substitute the value of $$y:$$
  $$8640-0.02	imes \frac{216000-2z}{11}+0.01z=8520$$
- step7: Multiply the terms:
  $$8640-\frac{108000-z}{275}+0.01z=8520$$
- step8: Convert the expressions:
  $$8640-\frac{108000-z}{275}+\frac{z}{100}=8520$$
- step9: Multiply both sides of the equation by LCD:
  $$\left(8640-\frac{108000-z}{275}+\frac{z}{100}ight)	imes 1100=8520	imes 1100$$
- step10: Simplify the equation:
  $$9072000+15z=9372000$$
- step11: Move the constant to the right side:
  $$15z=9372000-9072000$$
- step12: Subtract the numbers:
  $$15z=300000$$
- step13: Divide both sides:
  $$\frac{15z}{15}=\frac{300000}{15}$$
- step14: Divide the numbers:
  $$z=20000$$
- step15: Substitute the value of $$z:$$
  $$y=\frac{216000-2	imes 20000}{11}$$
- step16: Simplify:
  $$y=16000$$
- step17: Substitute the values of $$y,z:$$
  $$x=108000-16000-20000$$
- step18: Simplify:
  $$x=72000$$
- step19: Calculate:
  $$\left\{ \begin{array}{l}x=72000\y=16000\z=20000\end{array}ight.$$
- step20: Check the solution:
  $$\left\{ \begin{array}{l}x=72000\y=16000\z=20000\end{array}ight.$$
- step21: Rewrite:
  $$\left(x,y,zight) = \left(72000,16000,20000ight)$$
The amounts of the three parts of the investment are:
- First part: $$ \$72,000 $$
- Second part: $$ \$16,000 $$
- Third part: $$ \$20,000 $$
First part: $$ \$72,000 $$, Second part: $$ \$16,000 $$, Third part: $$ \$20,000 $$

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