Reflect on the concepts of linear and non-linear systems. What concepts (only the names) did you need to accommodate the concept of linear and non-linear systems in your mind? What are the simplest linear system and non-linear system you can imagine? In your day to day, is there any occurring fact that can be interpreted as linear systems and non-linear systems? What strategy are you using to get the graph of linear systems and non-linear systems?
Linear and non-linear systems are fundamental concepts in mathematics and have applications in various fields such as physics, engineering, economics, and computer science. These concepts require an understanding of certain terms and principles to accommodate them in our minds.
To comprehend linear systems, I must be familiar with terms such as linear equations, variables, coefficients, constants, and the concept of linearity. A linear system consists of linear equations, where each equation represents a straight line on a graph (Abramson, 2021). The simplest linear system can be represented by a single equation with two variables, such as y = mx + b, where m is the slope and b is the y-intercept. This equation represents a straight line on a graph.
On the other hand, non-linear systems involve non-linear equations, which do not represent straight lines on a graph. Non-linear systems can have various forms, including polynomial equations, exponential equations, logarithmic equations, and trigonometric equations (Abramson, 2021). The simplest non-linear system can be represented by a quadratic equation, such as y = ax^2 + bx + c, where a, b, and c are constants. This equation represents a parabola on a graph.
In our day-to-day lives, we encounter both linear and non-linear systems. Linear systems can be observed in scenarios where there is a direct relationship between two variables. For example, if we consider the relationship between the number of hours worked and the amount earned in a job, assuming a constant hourly wage rate, we can see that the more hours worked, the more money earned. This relationship can be represented by a linear equation. Another example can be calculating the cost of items based on their quantity or determining the distance traveled based on time and speed.
Non-linear systems are prevalent in situations where there is a complex relationship between variables. For instance, the growth of a population over time is typically modeled using non-linear equations. The rate of population growth is influenced by factors such as birth rate, death rate, immigration, and emigration, compound interest, or the spread of diseases which do not have a simple linear relationship.
To graph linear systems, the strategy typically involves identifying the slope and y-intercept of each equation and plotting the corresponding points on a graph. Connecting these points will result in a straight line that represents the linear system. The graph can be further analyzed to determine the solution to the system, which can be a single point of intersection, no intersection (parallel lines), or infinitely many intersections (coincident lines).
Non-linear systems, on the other hand, require more advanced techniques to graph. Depending on the type of non-linear equation, specific strategies may be employed. For example, quadratic equations can be graphed by finding the vertex, axis of symmetry, and a few additional points. Exponential and logarithmic equations may require plotting several points and observing the overall trend of the graph.
To sum up, understanding linear and non-linear systems requires familiarity with concepts such as linear equations, variables, coefficients, constants, linearity, and various types of non-linear equations. These concepts have real-life applications and can be observed in our day-to-day experiences. Graphing linear systems involves identifying the slope and y-intercept, while graphing non-linear systems may require more advanced techniques depending on the type of equation.
References
Abramson, J. (2021). Algebra and trigonometry (2nd ed.). OpenStax, TX: Rice University. Retrieved from https://openstax.org/details/books/algebra-and-trigonometry-2e