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System of equation: An Introduction with Methods and Examples

A system of equations is a collection of finite equations that are solved together to obtain a set of common solutions. Each equation in the system relates the variables in a particular way, and the aim is to find the values for the variables that fulfill all the equations.

A system of equations is applied in various everyday situations to model problems where the values that are not known can be expressed using variables. There are various methods available to find the solution to a system of linear equations. These methods include substitution, cross-multiplication, elimination, etc.

In this article, we will discuss a system of equations and its types. We will learn how to determine a system of equations. After this, we will discuss some examples with its systematic solution.

System of Equations: Definition with Example

In algebra, a group of mathematical equations that are solved together to find common solutions is called a system of equations. A system of equations is also called a set of simultaneous equations. Usually, we deal with two or three equations and variables in a system of equations. For example,

8x + 4y = 4

2x – 8y = 5

In the given system, two equations, and two variables x and y involves. The aim is to determine the value of x and y that fulfill both equations of the system simultaneously.

Note: For a system of equations, a solution that satisfies all the equations in the system is considered valid.

Types of systems of equations

Systems of equations can be classified as:

• Linear systems
• Nonlinear systems
• Undetermined systems
• Overdetermined systems
• Consistent systems
• Inconsistent systems
• Dependent systems
• Independent systems

Let’s explore the above types one by one.

Linear systems

Linear systems consist of equations where each variable has a power of one, and no other functions or exponents are involved. For example

Nonlinear systems

These are systems of equations where at least one equation is nonlinear, meaning that it involves an exponent, a logarithm, a trigonometric function, or some other non-linear function. For example,

Undetermined systems

Undetermined systems involve more variables than equations.

Overdetermined systems

These are systems of equations where there are more equations than variables. Like

Consistent systems

Systems of equations in which at least one solution exists are known as consistent systems.

Inconsistent systems

If the solution of the system of equations does not exist, then the system is known as an inconsistent system.

Dependent systems

In this system, the equations are not independent, which means that at least one equation can be obtained by combining the others.

Independent systems

In this system, the equations are independent, meaning that none of the equations can be obtained by combining the others.

Methods to find the solution to the system of equations

To find a solution to a system of equations, here are the most common ones:

• Substitution method
• Elimination method
• Graphical method

The choice of method depends upon the type and complexity of the system of equations, as well as personal preference and familiarity with the different methods.

Substitution method

To find the solution to the system of equations by substitution method, follow the following steps:

1. Select one of the equations of the given system and solve it for one of the variables in terms of the other.
2. Substitute the expression obtained in step-1 into the other equation to obtain an equation in one variable.
3. Find the value of the single variable by solving the equation.
4. Substitute the value obtained in step-3 into either of the original equations to find the value of other variables.
5. Write the obtained solution as ordered pair of the form (x,y).

Elimination method

To find the solution to the system of equations by Elimination method, follow the following steps:

1. Write the equations in standard form, with the x and y terms on the same side and the constant on the other side.
2. Choose one variable (either x or y) to eliminate by multiplying one or both equations by a constant so that the coefficient of the chosen variable is the same in both equations.
3. Add or subtract the equations together to eliminate the chosen variable and find the value of the other variable.
4. Substitute the obtained value of a variable in another equation to find the value of the eliminated variable.
5. Check the solution by substituting the value of x and y into both original equations to ensure they satisfy both equations simultaneously.
6. Write the obtained solution as ordered pair of the form (x,y).

Graphical method

To find the solution to the system of equations by graphical method, follow the following steps:

1. Write the equation of the given system in the form of y. i.e. y = ax + b
2. Graph both lines on the same coordinate plane.
3. The solution to the system is the point where the two lines intersect.

Note: The system has no solution if the lines are parallel and do not intersect. If the two lines overlap then the system has infinitely many solutions.

How to solve system of equations?

There are several methods to calculate the unknown terms of system of equations. You can try a system of equations calculator to find the unknown terms with steps online or apply elimination method, substitution method, or graphical method to find the unknown terms manually.

Let’s solve some examples with different methods

Example 1.

Determine the following system of equations by graphical method.

2x + y = 2

3x + y = 3

Solution:

2x + y = 2 _____ (i)

3x + y = 3 _____ (ii)

Find the value of y in terms of x, from (i)

y = 2 – 2x

 x - 1 -2 0 1 2 y 4 6 2 0 -2

Now, find the value of y in terms of x, from (ii)

y = 3 – 3x

 x - 1 -2 0 1 2 y 6 9 3 0 -3

Graph both lines on the same coordinate plane,

As graph of y = 2 – 2x, and y = 3 – 3x intersect at (1, 0).

Therefore (1, 0) is a solution set of the given system.

Example.2

Solve the following system by elimination method.

2x + 4 y = 2

3x - 2y = 3

Solution:

2x + 4 y = 2 _____ (1)

3x - 2y = 3 ______ (2)

We want to eliminate the y variable. We will equate the coefficient of y. For this, equation (2) multiply by 2. We have,

6x - 4y = 6 ______ (3)

Now add equation (1) and (3)

2x + 4 y = 2

6x - 4y = 6

8x          = 8

Let’s solve the remaining expression for x,

x = 8 / 8 = 1

x = 1

Substitute x = 1 in equation (2)

3(1) - 2y = 3

3 – 2y = 3

-2y = 0

y = 0 / -2 =0

y = 0

(1, 0) is the solution to the given system.

Conclusion

In this article, we have provided a detailed discussion of the system of equation. We have covered its different types. We discussed various methods to solve the system of equations. We learned how to determine the solution to the system of equations. We solved different examples by different methods for readers.