How do you find the value of x in the system of equations where x * y = 3 and 3 * x + 3 * y = 5?

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Finding the Value of x in a System of Equations

To solve for the value of x in the given system of equations:

  1. Equation 1: x * y = 3
  2. Equation 2: 3 * x + 3 * y = 5

We will start by rearranging these equations. From Equation 1, we can express y in terms of x:

y = 3/x

Now, we substitute this expression for y into Equation 2:

3 * x + 3 * (3/x) = 5

This simplifies to:

3 * x + 9/x = 5

To eliminate the fraction, multiply every term by x (assuming x ≠ 0):

3 * x^2 + 9 = 5 * x

This can be rearranged into a standard quadratic format:

3 * x^2 - 5 * x + 9 = 0

Next, we can apply the quadratic formula to find x: 

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 3, b = -5, and c = 9. Plugging these values into the formula:

x = (5 ± √((-5)² - 4 * 3 * 9)) / (2 * 3)

This simplifies to:

x = (5 ± √(25 - 108)) / 6

Calculating the discriminant:

25 - 108 = -83

Since the discriminant is negative, it indicates that there are no real solutions for x in this system. The solutions exist in the complex number space. Therefore:

x = (5 ± √(-83)) / 6

The value of x can be expressed as:

x = (5 ± i√83) / 6

In conclusion, the system of equations does not have real solutions for x. Instead, x takes on complex values.

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