Based on the system of equations, the amount invested at 6% is $5,000, the amount invested at 7% is $10,000, and the amount invested at 8% is $15,000.
Total amount invested = $30,000
Let the amount invested at 6% = x
Let the amount invested at 7% = y
Let the amount invested at 8% = z
z = x + y
x + y + z = $30,000 ...Equation 1
0.06x + 0.07y + 0.08z = $2200 ...Equation 2
x + y = z ...Equation 3
With the third equation, express z in terms of x and y: z = x + y.
Substitute z = x + y into the first equation: x + y + (x + y) = 30,000
Simplify: 2x + 2y = 30,000
Divide both sides by 2: x + y = $15,000
Thus, x + y = 15,000
Therefore, the amount invested at 8% = $15,000
Substitute z = 15,000
0.06x + 0.07y + 0.08(15,000) = 2,200
0.06x + 0.07y + 1200 = 2,200
0.06x + 0.07y = 1000
Now we have a system of two equations:
x + y = $15,000
0.06x + 0.07y = $1000
Solving this system of equations will give us the values of x and y, which represent the amounts invested at 6% and 7%, respectively.
Multiply x + y = $15,000 by 0.06:
0.06x + 0.06y = 900
Subtract 0.06x + 0.06y = 900 from 0.06x + 0.07y = $1000:
0.06x + 0.07y = $1000
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0.06x + 0.06y = 900
0.01y = 100
y = $10,000
Substitute y = $10,000 in 0.06x + 0.07y = $1000:
0.06x + 0.07(10,000) = $1,000
0.06x + 700 = $1,000
0.06x = $300
x = $5,000
Thus, we cann conclude that the amount invested at 6% is $5,000, the amount invested at 7% is $10,000, and the amount invested at 8% is $15,000.