Explanation:
To solve the given equations, let's assign variables to each equation and solve them step by step.
Let's assume:
a = x
b = y
c = z
d = w
e = v
From the given equations:
a * b = 1 -> x * y = 1 (Equation 1)
b * c = 4 -> y * z = 4 (Equation 2)
c * d = 9 -> z * w = 9 (Equation 3)
d * e = 16 -> w * v = 16 (Equation 4)
e * a = 25 -> v * x = 25 (Equation 5)
Now, let's solve the equations using substitution:
From Equation 1 (x * y = 1), we can rewrite it as y = 1/x.
Substituting y in Equation 2, we get (1/x) * z = 4, which gives us z = 4x.
Substituting z in Equation 3, we have (4x) * w = 9, which gives us w = 9/(4x).
Substituting w in Equation 4, we get (9/(4x)) * v = 16, which simplifies to v = (16 * 4x)/9.
Finally, substituting v in Equation 5, we have [(16 * 4x)/9] * x = 25.
Simplifying the equation, we get:
(64x^2)/9 = 25.
To solve for x, we can cross multiply and solve the quadratic equation:
64x^2 = 225.
Dividing both sides by 64, we get:
x^2 = 225/64.
Taking the square root of both sides, we have:
x = ±(√(225/64)).
So, x = ±(15/8).
Now, substituting the values of x in the respective equations, we can find the values of y, z, w, and v.
For x = 15/8:
y = 1/(15/8) = 8/15
z = 4 * (15/8) = 30/4 = 15/2
w = 9/(4 * (15/8)) = 9/(30/8) = 9 * (8/30) = 12/5
v = (16 * 4 * (15/8))/9 = (60/2)/9 = 60/18 = 10/3
For x = -15/8:
y = 1/(-15/8) = -8/15
z = 4 * (-15/8) = -30/4 = -15/2
w = 9/(4 * (-15/8)) = 9/(-30/8) = -9 * (8/30) = -12/5
v = (16 * 4 * (-15/8))/9 = (-60/2)/9 = -60/18 = -10/3
Therefore, the possible solutions for the variables are:
x = 15/8, y = 8/15, z = 15/2, w = 12/5, v = 10/3
or
x = -15/8, y = -8/15, z = -15/2, w = -12/5, v = -10/3.
Note: The solution includes both positive and negative values for the variables.