Sketching a System of Two Linear Equations and Exploring Its Solution (-1, 3)

Sketching a System of Two Linear Equations and Exploring Its Solution (-1, 3)

==========================

Throughout mathematical exploration, one fascinating aspect lies in constructing systems of linear equations with specific solutions. In this article, we will focus on devising a system of two linear equations whose solution set is (-1, 3). We will delve into the intricacies of this solution and discuss its implications in various contexts.

Sketching a System of Two Linear Equations

Consider the following system of two linear equations:

  1. 3x + 4y = k1
  2. 5x + y = k2 (where k1 and k2 are constants)

This system can be solved using various methods, including substitution and elimination. To find a solution set (-1, 3), we can begin by substituting these values into the equations. Substituting x = -1 and y = 3 in equation 1, we get:

3(-1) + 4(3) = k1 ⇒ -3 + 12 = k1 ⇒ k1 = 9.

Similarly, substituting x = -1 and y = 3 in equation 2, we get:

5(-1) + 3 = k2 ⇒ -5 + 3 = k2 ⇒ k2 = -2.

Hence, the system of equations with the solution (-1, 3) becomes:

  1. 3x + 4y = 9 (with x = -1 and y = 3).
  2. 5x + y = -2 (with x = -1 and y = 3).

Discussion on the Solution (-1, 3)

The solution (-1, 3) obtained from the system of equations offers a unique set of values that satisfy both equations simultaneously. In various contexts, this solution might have practical implications. For instance, in real-world applications like engineering or economics, where linear equations represent various parameters or variables, the solution (-1, 3) could indicate the optimal conditions for balancing those parameters or achieving a desired outcome. Additionally, the existence of this specific solution highlights the versatility of linear equation systems in finding real-world solutions.

Implications of the Solution in Different Fields

In geometry, the solution (-1, 3) could represent the intersection point of two lines represented by these equations. In algebra, it could indicate a common point of satisfaction for two polynomial functions at specific conditions or limitations. In engineering, the solution could reflect a design requirement that satisfies both stress and deformation criteria simultaneously. In economics, it could represent a state where demand and supply balance each other effectively. The significance of this solution lies in its ability to provide insights across various fields, pointing to scenarios where things are most optimal or in balance. Through the exploration of linear equation systems like these, we can gain deeper insights into real-world phenomena and find solutions to complex problems.

  1. How does the system change if we alter the constants k1 and k2? What are the new solutions?
  2. Can we find other systems of linear equations with different solutions but similar characteristics? How would they differ from this system?
  3. What are some practical applications where this solution might be relevant? Can you provide real-world examples?