The world of mathematics has many types of equations that scientists, economists, statisticians, and others use to explain, predict, and analyze the universe around them. Several variables are connected in these equations so that one can influence or forecast the output of another. Linear equations are common in elementary mathematics, but non-linear equations dominate in higher mathematics and science.

To identify them, you must learn to distinguish between linear and nonlinear equations.

## What is an equation?

An equation is a statement of equality between two algebraic expressions containing constants and variables. The left-hand side (LHS) and right-hand side (RHS) of the equation are the two sides of the equality sign.

For example, 5x + 10 = 35 where 5,10 and 35 are constants and x is a variable.

To solve an equation, we do a set of mathematical operations on both sides of the equation, such that the unknown variable is on one side and its value is on the other.

You may have come across different types of equations while solving mathematical problems. Few equations can have only numbers, while others can have only variables, while some can have both numbers and variables. They are classified as linear or non-linear equations based on the degree and variable in the equations.

Linear and non-linear equations can have both numbers and variables. In a graph, a linear equation represents a straight line, whereas non-linear equations indicate curves. Before we get into the differences between linear and non-linear equations, let us first define linear and non-linear equations.

## What is Linear Equation?

The word ‘Line’ in Linear represents a straight line. Something related to a line is referred to as linear. A line is constructed using all of the linear equations. Lines in the arranged framework represent these conditions. A linear equation is an equation describing a straight line.

The general representation of the linear equation is y = mx + c, where x and y are variables, m is the slope, and c is the constant. Linear Equations have a maximum of one degree.

Ex: x + 1 = 5

y-10 = 20

### Linear Equation with two variables

For ex: 3x + 4y = 10

y = 5x + 8

## What is a non-linear Equation?

The word “Non-Line” in Non-Linear means it is not a straight line but a curved line.

The general representation of the Non-Linear Equation is ax2 + by2 = c where x and y are variables and c is constant. A non-linear equation has a maximum degree of 2 or more than 2.

A few examples of Non-Linear Equations are

y = x2 + 5

3×2 + 4y2 = 10

y = x

6x + 7y = 14

## Difference between Linear and non-Linear Equations

The main difference between linear and non-linear equations is explained here so that students can grasp it more naturally. It is presented in tabular format with examples to make it easier for students to understand.

 LINEAR EQUATION NON-LINEAR EQUATION 1 Linear Equation represents a straight line on the graph Non-Linear Equation represents a curved line on the graph 2 A Linear equation can be defined as the equation having a maximum of only one degree. If an equation has a degree greater than or equal to 2, it is nonlinear. 3 A linear equation is of the form:  y = mx + c With x and y as variables, m represents the slope, and c is a constant Non-linear equation is of the form: ax2 + by2 = c Where x and y are variables and a,b, and c are constants. 4 In the XY plane, these equations form a straight line. This line can be extended in any direction. The graph forms a curve, and the curve of the graph increases as the degree is increased. 5 The slope value is constant. The slope is variable. 6 Ex  5x = 35  8y + 3x = 10  5x = 7y  100x + 34 = 12y Ex 5x2 + 4y2  = 9 a2 + 2ab + b2  = 0 x2 + x + 2 = 24

## HOW TO IDENTIFY LINEAR AND NON-LINEAR EQUATIONS

There are many methods to differentiate between the two:

1. Graphing,
2. Solving an equation and
3. Making a table of values.
1. GRAPHING
• If you were not given a graph, plot the equation as a graph.
•  Find out whether the line is straight or curved.
•  The equation is linear if the line is straight. It is a nonlinear equation if it is curved.
1. SOLVING AN EQUATION
•  Making Use of an Equation
•   Reduce the equation as much as feasible to the form y = mx + b.
•   Examine your equation to check whether it contains exponents. It is non-linear if it includes equations.
•   Your equation is linear if it contains no exponents. The slope is represented by “m.”
•   Check your work by graphing the equation. It is linear if it is a straight line. It is non-linear if the line is curved.
1. MAKE USE OF A TABLE
• Make a table with the sample x values and solve for the corresponding y values. Pick x values that are at a consistent distance from each other. Solve for y for x values of -4, -2, 2, and 4.
• Calculate the difference between the y values.
• If the difference is constant or has the same value, the equation is linear, and its slope is constant. If the differences are not equal, then the equation is not linear.

Let me explain it to you with a few examples. You will understand it better.

The variable x must be either zero or one degree, and variable y must be one degree to be a linear equation.

### For example:

1. y = 2x -3

Both x and y are first degrees.

2. 4x + 5y = 20

Both x and y are first degrees

3. 2x-4y = 7 + 3x

Here all variables are first degree

4. y = -1

Here x is degree zero, and y is the first degree

The result is a horizontal line that is a function of x If variable x is first degree, but variable y has zero degrees, it will be a linear relation but not a function of x.

### For example

1. x = 4

Graphs such as this are vertical lines and not a function of x.

If variable y is first degree, but variable x has a degree other than zero or one, it will be a non-linear function.

### For example

1. y = x2 + 25

x is not the first degree.

2. y = 5x + 2 -x3

x is third degree.

3. y = 1 /x

x is to the power of -1.

4. y = x

x is to the power ½. Here the graph is half a sideways parabola(curve).

5. y = 2x

x is the exponent here instead of the base. So the graph is exponential and not linear.

If variable y is not the first degree, the relation will not be a function of x

For example

1. x2 + y2 = 4

Neither x nor y is first degree. The graph will be a   circle with a radius of 2.

2. x = y2

Y is not the first degree. This is a sideway parabola(curve).

You should now be able to identify linear and non-linear equations.

If the maximum degree is one in an equation, then it is a linear equation, and if the variable is a square, square-root, exponential, or a fraction, then it is a non-linear equation. You can also test if the equation is linear or non-linear by plotting it on the graph. If the graph is a straight line, then it is a linear equation, and if the graph is curved, it is a non-linear equation.

## Conclusion

As soon as you look at the graph, you should identify whether it is linear or non-linear. The simplest way to remember and identify is to keep in mind that a straight line indicates linear, and anything other than a straight line will be non-linear. Simplifying math in this way will help you learn it well.

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