Difference between the Linear equations and Non-linear equations

In the field of discrete mathematics, we have already learned about the different types of equations. But in this section, we will learn about the difference between linear and non-linear equations. First, we will learn about the expression, and then we will learn about the linear and non-linear equations. Only after that, we can clearly understand the difference between the linear and non-linear equations.

Equations:

An equation can be described as a statement of the equality of two expressions. As we know that the equality sign always contains two sides, i.e., LHS and RHS. So the equations will also contain the two sides, i.e., LHS and RHS. At the time of solving the math problems, we have gone through many equations. Some equations can only contain the numbers, and some can only contain the variables, while all the other equations can have the numbers and variables both. The numbers and variables both are contained by the linear and non-linear equations. In any equation, we can determine whether the given equation is linear or non-linear with the help of calculating its degree and variable.

For example: Suppose there is an equation 3y + 6 = 10. Here y is used to indicate the variable, and 3, 6, and 10 are used to indicate the constants. The LHS (left-hand side) is provided by the expression 3y + 6, and the RHS (right-hand side) is provided by the constant 10. If we perform the same operation on the LHS and RHS of the equation, then the equation will remains unchanged.

We can solve linear and non-linear equations by using both sides of the equations and performing a series of identical mathematical operations on them. We will do this in such a way that an unknown variable will remain on one side, and the value of this variable will be obtained from another side. Now we will learn about linear and non-linear equations like this.

Linear Equation

The meaning of linear means a straight line. An equation will be known as a linear equation if it contains only a 1-degree term. With the help of values of linear equations, we can draw a straight line on the graph. In the following way, we can express the linear equation:

1. ax + b = c, where a is used to indicate the coefficient, b, c is used to indicate the constants, and x is used to indicate the variable. In this equation, there is only one variable. This form is the standard form of the linear equation for one variable.
2. There is one more form in which we can indicate the linear equation, which is described as follows: ax + by = c, where x and y are used to indicate the variables, c is used to indicate the constant, and a and b are used to indicate the coefficient. This type of equation is known as linear equation for two variables.

For example:

1. + 9 = 20, 6 /3y - 5 = 1, y² + 3 = 7, and y /5 + 7 = y /7 - 9

All the above equations contain only one variable, y. In all the above equations, each equation has the highest power as one. So, all the above equations are linear equations.

2. 4x + 5y = 19, 5x - y/2 = 2

Both the above equations contain the two variables x and y. In both equations, each equation has the highest power as one. So these equations are the second form of linear equation. So both the above equations are linear equations.

If we draw a graph with the help of a linear equation, then it will always form a straight line.

Non-linear Equations

An equation will be known as the non-linear equation if it contains 2 or more than 2-degree terms. The non-linear equation is a kind of equation that will not form a straight line. In any graph, it will be drawn in the form of a curve, and it will also contain a variable slope value. In the following way, we can express the non-linear equation:

ax² + by² = c, where a, b, and c are used to indicate the constants, and x, y are used to indicate the variables. In this equation, we have a minimum of 2 degrees. This form is the standard form of a non-linear equation.

For example:

1. 4x² + 5x + 3 = 0, y² - x = 7

All the above equations contain a minimum of two degrees. So, all the above equations are non-linear equations.

2. y = x² -7, √y + x = 6

Both the above equations contain the two variables x and y. The first equation contains degree 2, and the second equation contains a variable y whose power is ½. So both the above equations are non-linear equations.

If we draw a graph with the help of a non-linear equation, then it will always form a curve.

Difference between the Linear and Non-linear equations

There are many differences between linear and non-linear equations, and we will explain them with some examples. For this, we also should know the definition of linear and non-linear equations so that we can properly know the difference between them. In the following table, the differences between them are described as follows:

Examples of Linear and Non-linear equations:

There are various examples of linear and non-linear equations, and some of them are described as follows:

Example 1: In this example, we have a linear equation 9(x + 1) = 2(3x + 8), and we have to solve this.

Solution: From the question, we have a equation, 9(x + 1) = 2(3x + 8)

Now we will expand each side of this equation like this:

9x+9 = 6x+16

Now we will subtract 6x from both sides of the above equation like this:

9x+9-6x = 6x+16-6x

3x+9 = 16

Now we will subtract 9 from both sides of the above equation like this:

3x+9-9 = 16 - 9

3x = 7

Now we will divide each side by 3 like this:

3x /3 = 7/3

x = 7/3

Example 2: In this example, we have a non-linear equation 3x²-5x+2 = 0, and we have to solve this.

Solution: From the question, we have an equation, 3x²-5x+2 = 0

Now we will do the factorization of this equation:

3x² -3x-2x +2 = 0

3x(x-1) - 2(x-1) = 0

(3x-2) (x-1) = 0

(3x-2) = 0 or (x-1) = 0

x = 2/3 or x = 1

Example 3: In this example, we will solve an equation 3x + 9 = 2x + 18.

Solution: From the question, we have a equation, 3x + 9 = 2x + 18

⇒ 3x-2x = 18-9

⇒ x = 9

Example 4: In this example, we will solve the equations x+2y = 1 and x = y.

Solution: From the question, we have two equations,

x+2y = 1 …… (1)

x = y ……… (2)

Now we will put the value of equation 2 into equation 1 and get the following:

⇒ y+2y = 1

⇒ 3y = 1

⇒ y = 1/3

∴ x = y = 1/3

Example 5: In this example, we will solve the equation x = 12(x+2).

Solution: From the question, we has a equation,

x = 12(x+2)

x = 12x+24

Now we will subtract 24 from each side of the above equation like this:

x - 24= 12x+24-24

x - 24 = 12x

Now we will simplify the above equation like this:

11x = -24

Now we will divide each side of this equation by 11 and isolate x like this:

11x /11 = -24 /11

x = -24 /11

Example 6: In this example, we will solve the equations 7x+21 = 6x+26.

Solution: From the question, we have an equation,

7x+21 = 6x+26

7x-6x = 26-21

x = 5