Solving The System Of Equations 11x² + Y² = 132 And Xy = 11

In this article, we will delve into solving a system of equations using a method of our choice. The given system involves a quadratic equation and a simple equation, making it an interesting problem to tackle. Specifically, we aim to find the solution set for the following equations:

$$\begin{aligned} 11x^2 + y^2 &= 132 \\ xy &= 11 \end{aligned}$$

Solving such systems requires a blend of algebraic manipulation and strategic substitution. Let's explore the steps involved in finding the solution set.

Method Selection and Strategy

To solve this system, we can use the substitution method. This method involves solving one equation for one variable and substituting that expression into the other equation. The goal is to reduce the system into a single equation with one variable, which can then be solved using standard algebraic techniques. Given the equations:

$$\begin{aligned} 11x^2 + y^2 &= 132 \\ xy &= 11 \end{aligned}$$

The second equation, xy = 11, is easier to manipulate. We can solve for y in terms of x (or vice versa) and then substitute this expression into the first equation. This will give us an equation in terms of a single variable, which we can solve.

Step-by-Step Solution

1. Solve the second equation for y: xy = 11

   y = 11/x

2. Substitute the expression for y into the first equation: 11x² + y² = 132

   11x² + (11/x)² = 132

3. Simplify the equation: 11x² + 121/x² = 132

4. Multiply the entire equation by x² to eliminate the fraction: 11x⁴ + 121 = 132x²

5. Rearrange the equation into a quadratic form by subtracting 132x² from both sides: 11x⁴ - 132x² + 121 = 0

6. Divide the entire equation by 11 to simplify: x⁴ - 12x² + 11 = 0

Now, we have a quadratic-like equation in terms of x². Let u = x². The equation becomes:

u² - 12u + 11 = 0

This is a standard quadratic equation that can be factored or solved using the quadratic formula.

Solving the Quadratic Equation

To solve the quadratic equation u² - 12u + 11 = 0, we can factor it. We are looking for two numbers that multiply to 11 and add up to -12. These numbers are -11 and -1. Thus, we can factor the equation as:

(u - 11)(u - 1) = 0

Setting each factor equal to zero gives us the solutions for u:

• u - 11 = 0 => u = 11
• u - 1 = 0 => u = 1

Now, we need to substitute back x² for u to find the values of x.

Finding the Values of x

We have two values for u: u = 11 and u = 1. Substituting x² for u, we get:

1. x² = 11

   Taking the square root of both sides, we get:

   x = ±√11

2. x² = 1

   Taking the square root of both sides, we get:

   x = ±1

So, the values of x are x = √11, x = -√11, x = 1, and x = -1.

Finding the Values of y

Now that we have the values of x, we can find the corresponding values of y using the equation y = 11/x.

1. For x = √11:

   y = 11/√11 = √11

2. For x = -√11:

   y = 11/(-√11) = -√11

3. For x = 1:

   y = 11/1 = 11

4. For x = -1:

   y = 11/(-1) = -11

Thus, we have the following pairs of solutions (x, y):

• (√11, √11)
• (-√11, -√11)
• (1, 11)
• (-1, -11)

Solution Set

The solution set for the system of equations is the set of all ordered pairs (x, y) that satisfy both equations. From our calculations, we have found four solution pairs:

$$\{(\sqrt{11}, \sqrt{11}), (-\sqrt{11}, -\sqrt{11}), (1, 11), (-1, -11)\}$$

Therefore, the solution set is {(√11, √11), (-√11, -√11), (1, 11), (-1, -11)}.

Verification

To ensure our solution is correct, we can substitute each pair (x, y) back into the original equations and verify that they hold true.

1. For (x, y) = (√11, √11):

   • 11x² + y² = 11(11) + 11 = 121 + 11 = 132
   • xy = (√11)(√11) = 11

2. For (x, y) = (-√11, -√11):

   • 11x² + y² = 11(11) + 11 = 121 + 11 = 132
   • xy = (-√11)(-√11) = 11

3. For (x, y) = (1, 11):

   • 11x² + y² = 11(1)² + 11² = 11 + 121 = 132
   • xy = (1)(11) = 11

4. For (x, y) = (-1, -11):

   • 11x² + y² = 11(-1)² + (-11)² = 11 + 121 = 132
   • xy = (-1)(-11) = 11

All four pairs satisfy both equations, confirming that our solution set is correct.

Conclusion

In this article, we successfully solved the system of equations:

$$\begin{aligned} 11x^2 + y^2 &= 132 \\ xy &= 11 \end{aligned}$$

using the substitution method. By solving the second equation for y, substituting into the first equation, and simplifying, we found four solutions: (√11, √11), (-√11, -√11), (1, 11), and (-1, -11). We verified these solutions by substituting them back into the original equations. This exercise demonstrates a practical application of algebraic techniques in solving systems of equations. Understanding these methods is crucial for more advanced mathematical problems. Mastering the substitution method and recognizing quadratic forms are key skills in algebra.