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# Graphing Non Linear Equations

An equation which have variables with two or more degree is known as non linear equation. that is the non linear equation will contain terms like x2 , xy, y3, etc....In non linear system of equations one equation is linear and another one is non linear, here substitution is the easiest method to solve the problem. In the system of non linear equation, the linear equation is to be solved for either x or y, then substitute the resulting expression into the nonlinear equation. A non linear equation can be represented geometrically by a line whose points make up the collection of solutions of the equation.

Before we draw the graph of non-linear equation: we have to know about , What is equation ? what is linear equation? and What is non-linear equation?

So, What is Equation?

Answer: An equations is a expression of one or more terms separated with equal to "=" symbol. Terms can be numerical,alphanumerical, expression etc.

Examples: 1.) 7x – 2y = 21 (Linear equation)

2.) 4x2 + 6x - 5 = 0. (Non-Linear equation)

Now, What is Linear equation ?

A linear equation is very similiar to an algebraic equation of the from.

y = mx + b.

Where m is slope , And b is y-intercept.

Degree of the equation is one.It does not contain non linear terms.

Example : 1.) y = 8x - 9

2.) y = - x - 13.

Now, What is non-Linear equation?

Equation whose graph does not form a straight line (linear) is called a Nonlinear Equation. In a nonlinear equation, the equations are either of degree greater than 1 or less than 1, but never equal to 1.It contains non linear terms.

Examples: 1.) 4x2 + 2y - 1 = 0

2.)x3 + 2x2 - 4xy - 1 = 0

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## How to Graph Non Linear Equations

Step 1: If the given functions is in the format of f(x) = anxn + an-1xn-1 + an-2xn-2 + …. + a1x + a0

Where a0, a1.., an are stables.

Step 2: Now, Consider functions f(x) as y.

Step 3: Replace the various values for x = …-4, -3, -2, -1, 0, 1, 2, 3, 4…to obtain the

equivalent value for y.

Step 4: Insert all the x values and its equivalent y values in the chart.

Step 5: Make a point x and y – axis in a graph.

Step 6: Plot every points in the table.

Step 7: At last connect all the points which were plotted.

Problem 1) Solve the non –linear functions f(x) = x2 – x + 1

Step 1: The given functions is f(x) = x2 - x+1.

Step 2: Consider f(x) = y.

Step 3: Rewrite the given function as y= x2 - x+1.

Step 4: Now replace various values for x to get the equivalent value of y.

Consider x = -2

Substitute in given equation y= x2 - x+1.

then y = (-2) 2 - (-2) +1

y = 4 + 2 + 1

y = 7

Consider x = -1, substitute in given equation

then y= (-1) 2 - (-1) +1

y=1+1+1

y = 1+1

y=2.

Consider x = 0

then y= (0) 2 - (0) + 1

y=0 – 0 + 1

y =0 + 1

y=1.

Consider x = 1

then y= (1) 2 - (1) + 1

y=1 – 1 + 1

y = 0 + 1

y = 1.

Consider x = 2

then y= (2) 2 - (2) +1

y = 4 - 2 + 1

y = 2 +1

y = 3.

Step 5: Now, create a table for the given points

 X Y -2 7 -1 2 0 1 1 1 2 3

Step 6: Plot the points in a graph using the table

Step 7:Combine all the points.

This is the graph of non linear equation.

## Examples of Non Linear Equations

Problem : Solve the non linear system of equation and graph it.

y = x2

y = 32 – x2

Solution: Here the one equation is linear while the other equation is non linear

Plug first equation in equation two.

y = x2

x2 = 32 – x2

x2 + x2 = 32 – x2 + x2

2x2 = 32

Divide 2 on both sides

2x2/2 = 32/2

x2 = 16

x = 4, -4

Plug in x value in equation two

y = (4)2

y = 16

The co ordination points graphing non linear equations are (4, 16) and (-4, 16)

problem 2:

Solve the following non linear system of equation

y = 3x + 2

y = x2 + 2x – 3

Solution:

Here the one equation is linear while the other equation is non linear

Plug first equation in equation two.

y = 3x + 2

3x + 2 = x2 + 2x – 3

Subtract 3x and 2 on both sides

3x + 2 – 3x – 2 = x2 + 2x - 3 - 2 – 3x

x2 – x – 5 = 0

Here we have to use quadratic formula to solve this equation

x = (- b ± sqrt((b^2(2)-4ac)) ) / 2a

Here a = 1, b = -1, c = -5

x = (- (-1) ± sqrt((-2)2-4(1)(-5)) ) /2(1)

x = (1 ± sqrt(21) )/2

Square root of sqrt(21) = 4.58

x = (1+ 4.58)/2 , (1-4.58)/2

x = 5.48/2 , -3.58/2

(x1 , x2) = 2.74 , -1.79

Plug in the x value in the second equation

y1 = (2.74)2 + 2.74 – 3

y1 = (7.50) - 0.26

y1 = 7.24

y = (-1.79)2 + (-1.79) - 3

y = 3.20 - 4.79

y2 = -1.59

The co ordinate points graphing non linear equations are (2.74, 7.24) and (-1.79, -1.59)

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