91Ó°ÊÓ

In this exercise, we have a system of non-linear equations. Non-linear equations involve terms with variables raised to powers other than one or variables multiplied together, like in our equations:
\ \( x^2 + xy = 5 \) and \ \( 2x^2 + xy = 2 \).

Here are some key points to understand about non-linear equations:

- They can create curves when graphed, rather than straight lines.
- Solutions to these equations can be more complex compared to linear equations.
- Methods like substitution or elimination can still be used to solve them, but algebraic manipulation might become more involved.

By identifying and labeling these equations in the problem, it becomes easier to apply various techniques to find potential solutions.

Algebraic manipulation is essential when solving systems of equations. In this exercise, we use subtraction to eliminate one variable.

Here is a step-by-step breakdown:

• Label the equations for reference:
  Equation (1): \( x^2 + xy = 5 \)
  Equation (2): \( 2x^2 + xy = 2 \)


• Subtract Equation (1) from Equation (2) to eliminate \( xy \):
  \( (2x^2 + xy) - (x^2 + xy) = 2 - 5 \)


• This simplifies to: \( x^2 = -3 \)

Effective algebraic manipulation allows us to simplify a complex problem and reveal key insights about the solutions. In this case, it helped us determine the impossibility of having a real number solution.

To solve mathematical equations, we often seek real number solutions. Real numbers include both positive and negative integers, fractions, and irrational numbers.

For our system of equations, \ \( x^2 + xy = 5 \) and \ \( 2x^2 + xy = 2 \), algebraic manipulation led us to the equation \( x^2 = -3 \).

However, we need to keep in mind:

• The square of a real number is always non-negative.
  Hence, \( x^2 = -3 \) has no real solution.

Understanding why a solution does not exist is as important as finding one. Always check the feasibility of the solution within the set of real numbers. In this problem, it is clear that no real numbers satisfy the given system of equations.