a = 4, b = -0.25, c = 0.5
or a = -4, b = 0.25, c = -0.5
Write down what you know in the form of equations:
a / c = 8
a / b = -16
b / c = -0.5
a * c = 2
a * b = -1
Now see if you can express one variable in terms of another. The a * c = 2 expression looks like a nice place to begin.
a * c = 2
Divide both sides by c
a = 2/c
Now look for another expression that uses a and c. The expression a/c = 8 looks like a good possibility.
a/c = 8
Substitute the expression defining a in terms of c into the expression
(2/c)/c = 8
Multiply both sides by c
2/c = 8c
Multiply both sides by c again
2 = 8c^2
divide both sides by 8
2/8 = c^2
1/4 = c^2
Take the square root of both sides
1/2 = c
0.5 = c
Now that we know the value for c, Calculate a using the a = 2/c expression we made.
a = 2/c = 2/.5 = 4
So we now know that a = 4, and c = 0.5
Look for an expression using b that we can solve. a / b = -16 looks like a good place to try
a / b = -16
Substitute the known value of a
4 / b = -16
Multiply both sides by b
4 = -16 b
Divide both sides by -16
-4/16 = b
Simplify
-0.25 = b
So a = 4, b = -0.25, c = 0.5
Verify using the equations given
a / c = 8: 4 / 0.5 = 8 Correct.
a / b = -16: 4 / (-0.25) = -16 Correct
b / c = -0.5: -0.25 / 0.5 = -0.5 Correct
a * c = 2: 4 * 0.5 = 2 Correct
a * b = -1: 4 * -0.25 = -1 Correct
All of the quotients and products come out as they should, so the values determined for a,b,c are correct.
Note: Since the square root was taken for the equation
0.25 = c^2
0.5 = c
The value of either 0.5, or -0.5 is correct for c. So this problem actually has 2 solutions.
They are a = 4, b = -0.25, c = 0.5
or a = -4, b = 0.25, c = -0.5
