# Solve the system by the substitution method: xy = 12, x^{2} + y^{2} = 40.

Equations are very important and are used for calculating various quantities in the field of engineering and science. There are various methods in which an equation can be solved; one of them being a substitution.

## Answer: The solution to the system of equations xy = 12, x^{2} + y^{2} = 40 are x = 2, y = 6; x = -2, y = -6; x = 6, y = 2; x = -6, y = -2.

Let's understand the solution in detail.

**Explanation:**

We have to use the method of substitution to solve the problem.

We have equations:

⇒ xy = 12 ------(1)

⇒ x^{2} + y^{2} = 40 ------(2)

Hence, we write equation (1) as y in terms of x, i.e,

⇒ y = 12/x ------(3)

Now, we substitute the value of y from equation (3) to equation (2).

⇒ x^{2} + 144/x^{2} = 40

⇒ x^{4} - 40x^{2} + 144 = 0

Now, we solve for x in the above quadratic equation. Now let's take z = x^{2 } ------(4)

⇒ z^{2} - 40z + 144 = 0

We use the quadratic formula to solve the equation above. After solving the equation, we get z = 4 and z = 36.

Hence, we get x = -2, 2 from z = 4, and x = 6, -6 from z = 36.

Now, after we substitute these values in equation (1), we get the following values as our solutions:

⇒ x = 2, y = 6

⇒ x = -2, y = -6

⇒ x = 6, y = 2

⇒ x = -6, y = -2