Types of the system of equations

A system of equations consists of two or more linear equations that possess more than one variable. A solution to a system of equations is typically the set of values capable of solving the linear equations and satisfying the system of equations simultaneously. 

A system has two major properties:

1. It consists of several functions that interact with and impact each other.

2. It produces an output when an input is provided. 

Similarly, a system of equations is a linear system that consists of an input, which are the values of the variables, and various functions, which are the coefficients that impact each other. 

A linear system is where the output is dependent on the input. Similarly, a system of linear equations is also dependent on the input, i.e., the variables. 

Characteristics of systems of equations

A system of equations has suitable characteristics for the linear equations:

1. They can be represented in the form of a graph, i.e., fulfilling the graphical equation 

y = mx + b

where m = slope of the graph

and b = y-intercept 

1. There are no curves, only straight lines that arise from the linear equation in a system of linear equations.

2. Any point on one line is the solution to the variable of that equation, but the solution to two or more equations simultaneously needs to be the point where these lines intersect.

Solutions to a system of equations

The point at which the graphs of all the equations intersect is the solution to a system of linear equations. A linear system of equations can possess a unique solution, infinite solutions, or no solutions. Let us explore the different solutions of a system of equations:

1. Unique solutions: when the lines of the equations in a system of equations intersect at a single point and no other, the system is said to possess a single or unique solution. 

2. Infinite solutions: when the lines of the equations in a system of equations intersect at multiple points, which usually occurs when they have the same slope and same y-intercept, they are assumed to be the same line and hence overlap in many places. In such cases, they possess infinite solutions.

3. No solution: when the lines of the equations in a system of equations do not intersect at all, which usually occurs when they have the same slope but different y-intercept, they are said to have no solution. Such lines are parallel lines. 

Types of the system of equations

Based on the types of solutions a system of equations can have, the systems of equations are divided into two types:

1. Consistent system: A system with a unique or single solution is said to be consistent. This is often referred to as a “normal” system if it represents an actual physical system or real-life system.

2. Inconsistent system: A system that has infinite solutions is said to be inconsistent or redundant. This is because of the repetitive solutions that the system provides due to its nature. This system is considered an inaccurate analysis if used to represent a real-life system. 

Based on the determinants of the equations, the systems are divided into two types:

1. Homogenous

2. Non-homogenous

Let’s consider a system of equations as follows:

a1x + b1y + c1z = d1 

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3 

[Notice these equations are not shown in the y = mx + b format; however, they need to be rearranged into that format before plotting a graph to determine their solutions.]

If d1 = d2 = d3 = 0, then the system is said to be homogenous; otherwise, it is non-homogenous. 

How are systems of equations used in real life?

In real life, rather specifically in physics, systems of equations are typically used to represent equations of motion. These are equations that describe the behaviour of a physical or real-life system in terms of kinetics as a function of time. You will often find problems that focus on the comparative behaviour of two sets with a common physical function. These problems can be easily solved using a system of equations.

The most common examples of these problems involve velocity, upstream and downstream, or those that compare distances covered by two systems in different amounts of time.

Let’s take an example.

A boy can run at a speed of 0.2 km per minute. His friend can bicycle at a speed of 0.6 km per minute. However, the friend requires at least 5 minutes to oil his bicycle before he can start. How long will his friend take to overtake the boy? 

Let’s assume that the boy covers x distance in y time. 

We know that distance = speed * time

So, for the boy, we get x = 0.2 y

Similarly, for his friend, we get x = 0.6 (y – 5) [Note: 5 is subtracted from the time taken by the friend as the friend starts 5 minutes later due to the time required to oil his bicycle.]

Therefore, we get two different sets of equations with a unique solution as they represent a physical setup.

Conclusion

A system of equations is a collection of two or more equations with the same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. A system of linear equations is a very important concept, as it is used to represent and solve many real-life problems, including those of physics, engineering, and mathematics.