An investment of ​$54,000 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 $ 4260.  The interest from the first investment was 6 times the interest from the second. Find the amounts of the three parts of the investment.

x * 0.08 y* 0.09 z* 0.06 = 4260 I multiply by 100 (1)
8x+9y+6z=426,000 (1)
8x+6(9y)=0 (2)
8x+54y=0 (2)
x+y+z=54,000 (3)
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8x+9y+6z=426,000 (1)
x+y+z=54,000 (3) *-6
-6x-6y-6z=-324000 (4)
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8x+9y+6z=426,000 (1)
-6x-6y-6z=-324000 (4)
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2x+3y=102000 (5) *-4
-8x-12y=-408000
8x+54y=0 (2)
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42y=-408000
y= -68000/7 I find the y
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is that correct ?

how can y be negative?

x+y+z = 54000
.08x + .06y + .09z = 4260
.08x = 6*.06y

x,y,z = 36000, 8000, 10000
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So, where did you go wrong?

8x+6(9y)=0

Why do you think the interest sums to zero?

Your process seems correct up to this point. However, you made a slight mistake in your calculations. Let's recheck the steps together:

1. We have the equations:
- x * 0.08 + y * 0.09 + z * 0.06 = 4260 (equation 1)
- x + y + z = 54000 (equation 3)

2. To eliminate decimals, multiply equation 1 by 100:
- 8x + 9y + 6z = 426000 (equation 1, modified)

3. Rearrange equation 1 by subtracting 8x from both sides:
- 9y + 6z = 426000 - 8x

4. We are given that the interest from the first investment is 6 times the interest from the second. So we can write:
- 8x = 6 * (9y)
- 8x = 54y

5. Substitute the value of 8x from equation 2 into equation 1:
- 54y + 9y + 6z = 426000

6. Combine like terms:
- 63y + 6z = 426000

7. Simplify equation 3:
- x + y + z = 54000

8. Now we have the system of equations:
- 54y + 9y + 6z = 426000 (equation 1, modified)
- x + y + z = 54000 (equation 3)

9. Multiply equation 3 by -6:
-6x - 6y - 6z = -324000

10. Add equation 1 (modified) and equation 4 together to eliminate z:
-6x - 6y - 6z + 54y + 9y + 6z = -324000 + 426000

11. Simplify and combine like terms:
48y = 102000

12. Solve for y:
y = 102000 / 48
y = 2125

So, the value of y is 2125.

To find the values of x and z, substitute the value of y back into either equation 1 or 3. Let's use equation 3:

x + 2125 + z = 54000
x + z = 54000 - 2125
x + z = 51875

Now, we have two variables and one equation. However, their values are not unique. We need additional information to find the specific values of x and z.

No, your calculation of y seems to be incorrect. Let's go through the calculations step by step to find the correct value.

From equation (2):
8x + 54y = 0 (2)

Now, let's solve equation (5) and (2) simultaneously to find the values of x and y.

Multiplying equation (5) by -4:

-8x - 12y = -408,000

Adding equation (2) and (-8x - 12y = -408,000) together:

8x + 54y + (-8x - 12y) = 0 + (-408,000)

42y = -408,000

Dividing both sides by 42:

y = -408,000 / 42
y = -9,714.28

It looks like there may have been a mistake in the previous calculation. Now let's continue to find the values of x and z.

From equation (3):
x + y + z = 54,000

Substituting the calculated value of y:

x - 9,714.28 + z = 54,000

Rearranging the equation:

x + z = 54,000 + 9,714.28
x + z = 63,714.28

From equation (1):
x * 0.08 + y * 0.09 + z * 0.06 = 4,260

Substituting the values of y and rearranging the equation:

x * 0.08 + (-9,714.28) * 0.09 + z * 0.06 = 4,260

0.08x - 874.29 + 0.06z = 4,260

Rearranging the equation:

0.08x + 0.06z = 4,260 + 874.29
0.08x + 0.06z = 5,134.29

Now, we can solve the system of equations:

x + z = 63,714.28 (6)
0.08x + 0.06z = 5,134.29 (7)

Multiplying equation (7) by 100 to remove decimals:

8x + 6z = 513,429

Multiplying equation (6) by -8:

-8x - 8z = -509,714.24

Adding equations (6) and (-8x - 8z = -509,714.24) together:

-2z = 3,714.76

Dividing both sides by -2:

z = 3,714.76 / -2
z = -1,857.38

Substituting the value of z back into equation (6):

x + (-1,857.38) = 63,714.28

Rearranging the equation:

x = 63,714.28 + 1,857.38
x = 65,571.66

To summarize:
x = $65,571.66
y = $9,714.28
z = -$1,857.38

Therefore, the amounts of the three parts of the investment are approximately:
$65,571.66, $9,714.28, and -$1,857.38.