# An Introduction to System of Linear Equations with Examples

Linear equations play an utmost crucial role in the study of algebra especially in the area of linear algebra, mechanics, geometry, geology, astronomy, environmental science (to elaborate the concentration of pollutants in environment with the passage of time).

In real life context, linear equations also describe the relationship of different factors. In this article we will discuss briefly definition of system of linear equations, it’s types, some methods to solve system of linear equations with the help of some technical examples.

**Definition of System of Linear Equations?**

A system of linear equations / linear system is a finite set of linear equations of the form. A solution of a linear system is a sequence of n numbers which satisfy the relevant equation in the system. It is worthiness to note here that linear equations represent a straight line when these are plotted on a rectangular coordinate system.

The system of linear equations are:

Ax_{1} + Bx_{2} = C

Px_{1} + Qx_{2} = R

**Types of System of Linear Equations:**

Different types of linear system are as follows:

### Homogeneous Linear System:

If vector constant term is zero (as b in eq. 1.) on the right hand side of every equation in the given system of linear equations, then this system of linear equations is known as homogeneous linear system.

By theorem such kind of linear system will be one solution at least which is termed as its trivial solution.

### Non-homogeneous Linear System:

If vector constant is not equal to zero on the right hand side of every equation in the given system of linear equations i.e. b ≠ 0 in (1), then the system of linear equations is known as non-homogeneous linear system.

### Dependent Linear System:

A system of linear equations which has infinite solution is categorized as dependent linear system. In this system of linear equations, their graphs coincide i.e. equations show identical graphs (lines). In this linear system slopes and intercepts for the equations are same.

### Independent Linear System:

A system of linear equations which has exactly one solution is categorized as independent linear system. In this type of linear system equations lines intersect (coincide) at only one point and their graphs are not parallel.

### Consistent Linear System:

A system of linear equations that has at least one solution is known as consistent linear system.

### Inconsistent Linear System:

A system of linear equations that has no solution is said to inconsistent. In this type of linear system graph of the equations never meet at a point i.e. their graphs are parallel.

**Methods to Solve Linear System:**

Here we will discuss two most of the important and precise methods.

### Elimination Method:

The elimination method normally is used when multiples of one variable (coefficients) in both equations are identical. In this method of the solution of the linear system, we add or subtract both equations depending upon the situation.

If variables of equal coefficients have same sign, then we subtract one equation from second equation. If variables of same coefficients have opposite sign, we add both of the equations in this case.

**Example 1. **

Solve following system of linear equations by using elimination method.

3x – y = 12 …………… (1)

2x + y = 13 …………… (2)

**Solution: **

**Step 1.** Write down here the given system of linear equations.

3x – y = 12

2x + y = 13

**Step 2.** Add both equations to remove one variable.

3x – y = 12

2x + y = 13

5x + 0y = 25

5x = 25

x = 25/5

x = 5

**Step 3.**** **Place this value of x in equation (2) to get value of y.

2(5) + y = 13

10 + y = 13

y = 13 – 10

y = 3

Hence x = 5, y= 3.

**Verification:**

**Step 1.** Place these values of x and y in equation 1.

3(5) – 3 = 12

15 – 3 = 12

12 = 12 (True)

**Step 2. **Place these values of x and y in equation 2.

2(5) + 3 = 13

10 + 3 = 13

13 = 13 (True)

Since the values of x and y verifies both equations, hence we can conclude that it is the solution of the system of the linear equations.

- Substitution method:

In substitution method, we arrange one equation to express one unknown (variable x or y) in terms of the other variable. After then we place this expression of expressed variable into the other equation to get an equation in only one variable.

Then we solve this obtained equation to find out the value of variable existing in this equation. Now we try to understand the solution of the system of linear equations by substitution method.

**Example: Solve the following system of linear equation using substitution method.**

7x – 2y = 21 ………… (1)

4x + y = 57 ………….. (2)

**Solution: **

**Step 1.**** **Arrange equation (2)

y = 57 – 4x …………. (3)

**Step 2.** Place this expression of y in equation (1)

7x – 2(57 – 4x) = 21

**Step 3.** Simplify using different arithmetic operations.

7x – 114 + 8x = 21

7x + 8x = 21 + 114

15x = 135

x = 135/15

x = 9

**Step 4.** Place this value of x in equation (3).

y = 57 – 4(9)

y = 57 – 36

y = 21

**Verification: **

**Step 1.** Place these values of x and y in equation (1).

From equation (1)

7(9) – 2(21) = 21

63 – 42 = 21

21 = 21 (True)

**Step 2.** Place these values of x and y in equation (2).

4(9) + 21 = 57

36 + 21 = 57

57 = 57 (True)

Hence x = 9 and y = 21 is the solution of the system of the linear equation.

You can also use a system of equations calculator to find the unknown values of linear equations either with the elimination method or substitution method with steps.

**Wrap Up:**

In this article we talked over system of linear equations briefly. We elaborated system of linear equations and we discussed some of the famous types of system of linear equations. We’ve also elaborated two methods for to solve the system of linear equations with the help of examples.