Systems of Non Linear Equations

Algebraic equations are normally divided in to two major types they are Linear equations and nonlinear equations. Linear equation has the power value as 1. But non linear equation’s variable has the power value of more than 1.Linear equation produce a straight line on the graph. But the non linear equation does not produce straight line.

Solutions to equations are the points that made the equation true that made the equation work correctly. Normally, non linear algebraic equation problems are frequently accurately solvable, and if not they generally can be thoroughly understand through qualitative and numerical analysis.

Procedure for systems of non linear equations:

The numerical methods for estimate of real solutions of a system of non linear equations, i.e., finding the roots of a vector function. When compared with the one-dimensional case, finding roots in the multidimensional case is much more complex. For example, in one-dimensional case one can comparatively easily bracket the roots of a given function (i.e., conclude the intervals, in which at least one root of the function lies) but there are no methods for bracketing of roots of general functions in the multidimensional case! Usually we even do not know whether a root (solution of the system) exists and whether it is unique.

Example for systems of non linear equations:

Solve the following systems non linear equation:

y = x2

y = 18 – x2

Solution:

y = x2         ———–equation 1

y = 18 – x2  ———– equation 2

y = y

Plugging equation 1 in equation 2,

x2 = 18 – x2

x2 + x2 = 18 – x2 + x2

Each of these sub-equations is true, but  the last one is usefully new and different:

Solve this for the x-values that make the equation true:

x2 = 18 – x2

2x2 = 18

x2 = 9

x = –3, +3

Then the solutions to the unique system will occur when x = –3 and when x = +3.

Plug the x-values in to either of the two original equations. plug the x-values into the first equation, since it’s the simpler of the two:

x = –3:

y = x2

y = (–3)2 = 9

x = +3:

y = x2

y = (+3)2 = 9

Then the solutions are (x, y) = (–3, 9) and (3, 9).

Writing Algebra Equations

In Algebra, an equation is the basic number “sentence”. A mathematical expression which contain an equals sign is called as equation. If two algebraic expressions are equated, we get what are called algebraic equations. An algebra equation is a mathematical symbols which contains an equals sign. It says that two expressions mean the same thing, or which represent the same number.

An algebra equation can contain variables and constants. We can show math facts in short, easy-to-remember forms and solve problems quickly using equation.

Examples for algebra equation

Example 1:

2x + 2 = 14

Solution:

Subtracting by 2 on both sides,

2x + 2 – 2 = 14 – 2

2x = 12

Dividing by 2 on both sides,

2x / 2 = 12 / 2

x = 12 / 2

x = 6

Example 2:

x – 7 = 20

Solution:

Adding 7 on both sides,

x – 7 + 7 = 20 + 7

x = 20 + 7

x = 27

Example 3:

p + 2p = 6

Solution:

3p = 6

Diving by 3 on both sides,

3p / 3 = 6 / 3

p = 6 / 3

p = 2

Word Problems:

The most important skill that has to be developed in algebra is the ability to translate a word problem into the correct equation, so that you can solve the problem easily.

Let’s try some examples:

Example 1:

A number n times 3 is equal to 120.

This is an easy one. The word ‘times’ says you that you must multiply the variable n by 3, and that the result is equal to 120. Here’s how to write this equation:

3n = 120

Example 2:

Grey worked for 8 hours on Thursday and mowed 4 lawns. How much time, on average, did he spend on each lawn?

Solution:

Let the letter “a” represent the average time per lawn, the unknown value. Then, 4a would represent the time to mow all three lawns, and we know that this is equal to 8 hours. We can write the equation like this:

4a = 8 hours

a = 8/4

a = 2

Hence the average Time = 2 hours

Sum of Polynomials

Polynomial comes from poly- (meaning “many”) and -nominal (in this case meaning “term”) .  so it says “many terms” A polynomial is a monomial or sum of monomials. Some polynomials have special names. A binomial is the sum of two monomials. And a trinomial is the sum of three monomials. Polynomials with more than three terms have no special names.  Polynomials can be used to express geometric relationships. Monomials such as 5x and -3x are called like terms because they have the same variable to same power. When you use algebra tiles, you can recognize like terms because the individual tiles have the same size and shapes.

Adding polynomials:

Each expression is a polynomial. If it is a polynomial identify the type of the polynomial such as a monomial, binomial or trinomial. A polynomial can have constants, variables and exponents. To add polynomials, you can group like terms horizontally or write them in column form, aligning like terms.

Steps involving the addition of polynomials:

Step 1: Write the given polynomials

Step 2: combine like terms and remove zero pairs.

Step 3: Add the like terms

Example1:

Add     5x2 + 6x + 5     and     2x2 - 2x – 1

Given: 5x2 + 6x + 5     +     2x2 - 2x – 1
Place like terms : 5x2 + 2x2 + 6x – 2x + 5 – 1
Add : (5+2)x2 + (6-2)x + (5-1)

 

Then we can get 7 x+4 x +4

Example 2:

Add     (12x2 + 15y + 13xy),   (13x2 – 10xy – 15x)   and   (16xy + 10)

Line them up in columns and add:

12x2 + 15y + 13xy
13x2      - 10xy – 15x
16xy     + 10

25x2 + 15y + 19xy – 15x + 10

So the answer is 25x2 + 15y + 19xy – 15x + 10

This is the method of adding polynomials.

Proper Rational Expression

A rational expression is nothing more than a fraction in the numerator and/or the denominator are polynomials. Here are rational expression is expressed in the polynomial functions.

     F(x)=P(x)/Q(x)

Where P and Q are polynomial functions in and Q is not the zero polynomial. The domain F is the set of all points x in which the denominator Q(x)is not zero, assumes that the fraction is written in its lower degree, that is, P and Q have no common factor of positive degree.

Every polynomial function is a rational function Q(x) = 1. A function that cannot be written in the form (for example, f(x) = sin(x)) is not a rational function.

An expression of the form P(x)/Q(x) is called a rational expression.

Application of rational expression:

Rational functions used in numerical analysis for interpolation and of functions, for example the Padé approximations introduced by Henri Padé. In terms of rational functions are well suited in computer algebra systems and other numerical software. Like polynomials, they can be evaluated straightforwardly.

Rational functions are used model more complex equations in science and engineering including (i) fields and forces in physics, (ii) spectroscopy in analytical chemistry, (iii) enzyme kinetics in biochemistry, (iv) electronic circuitry, (v) aerodynamics, (vi) medicine concentrations in vivo, (vii) wave functions for atoms and molecules, (viii) optics and photography to improve image.

Find the domain of 3/x.

The domain is all values that x is allowed. Since I can’t divide by zero, I need to find all values of x that would be division by zero. The domain will be all other x-values. When is this denominator equal to zero? When x = 0.

Then the domain is “all x not equal to zero”.

Determine the domain of x/3.

The domain doesn’t care in the numerator of a rational expression. The domain is only influenced by zeroes of the denominator. Will “3″ ever equal zero?. Since the denominator will never equal zero, then there are no forbidden values for this rational expression, and x can be anything. So the domain is “all x”.

Problem

To find the domain ignore the “x + 2″ in the numerator and instead I’ll look at the denominator. I’ll set the denominator equal to zero, and solve. The x-values in the solution will be the x-values which would cause division by zero. The domain will then be all other x-values.

x2 + 2x – 15 = 0

(x + 5)(x – 3) = 0

x = –5, x = 3

x not equal to -5,3.

Degree of polynomial

The degree of polynomial is the greatest exponent of a term. The greatest exponent should have a non-zero coefficient in  a polynomial expressed as a sum or difference of terms which is commonly known as Canonical form. The sum of the powers of all variables in the term is the degree of a polynomial. The degree can also be specified as order. The degree of polynomial is for the single variable or the combination of two or more variables with the powers.

Properties of degree of polynomial

According to the degree of polynomials the names are assigned. Below listed are the degree of polynomials:

  • The name of the zero degree polynomial is constant.
  • The name of the 1 degree polynomial is linear.
  • The name of the 2 degree polynomial is quadratic.
  • The name of the 3 degree polynomial is cubic.
  • The name of the 4 degree polynomial is quartic
  • The name of the 5 degree polynomial is quintic.
  • The multiplicative inverse 1/a have degree -1
  • The square root has a degree as ½
  • The logarithm has a degree as 0, example log b.
  • The exponential function’s degree is infinity.

How to find the Degree of polynomial

In this section, learn how to find the degree of polynomial -

Using one variable:

Let us consider the polynomial, 8x^3+7x^4+6x^2+5x+1.

The degree of the first term is 3.

The degree of the second term is 4

The degree of the third term is 2

The degree of the fourth term is 1 and

The degree of the last term is 0.

Here the greatest degree among all the degrees is 4. So the degree of the polynomial

8x^3+7x^4+6x^2+5x+1 is 4.

 Using two variables:

Let us consider the given polynomial is 5x^2y^4+2xy^3+8y^2+2y+x^3+1

The degree of the first term is sum of the power of x and y. So the degree for the first term is 2+4 which is 6.

The degree of the second term is 1+3=4.

The degree of the third term is 2.

The degree of the fourth term is 1.

The degree of the fifth term is 3 and

The degree of the last term is 0.

Here the maximum degree is 4. So 4 is the degree of the polynomial 5x^2y^4+2xy^3+8y^2+2y+x^3

Linear Model Equation

Linear model equation is equation with degree one.

Degree of equation: Degree of any equation is highest power of variable in that equation, for example 2x2 + 3x + 5 = 0 for this equation highest power of variable is 2 so degree of this equation will be 2.

Variable: It can vary or change for example 2x + 5y = 3 here x and y is variable because they can change their values. It is generally represented by alphabetical letter.

For linear model equation power of all variables will be one.

5x + 6y = 3 in this equation powers of variable x and y are 1, so it’s a linear equation.

Linear equation can be with one variable, two variables or more.

ax + b = c for this equation a, b and c are constant and x is variable, for this equation we have only one variable so this is a linear equation with one variable

For this type of equation we will try to put variable at one side of equality and constants another side of equality, and then we can find value for variable.

Example: Solve the equation 2X + 6 = 8

Solution:   2X + 6 =8

Try to put variable at one side and put constants to another side.

2X + 6 = 8

-6   -6

2X = 2

X = 1   this is the solution of equation

Linear Model Equation with Two Variables:

Linear model equation with two variables will have two variables in equation.

For example 2X + 5Y = 6 for this equation we have two variable X and Y.

General equation for linear equation with two variables

ax + by + c = 0 here a,b,c are constants and x and y are variables.

Graph of these types of equations will be straight line

How to draw graph

To draw graph of linear model equation we will take minimum 3 value of X and try to find corresponding values of Y, then we will have three points and all will be collinear

When we will join these three points we will get graph of linear model equation.

Let’s take an example

4X + 3Y = 12

Step 1: Make table for X and corresponding Y

4X + 3Y = 12

3Y = 12- 4X

Y = 12-4X/3

X Y=12-4X/3
0 4
3/2 2
3 0

 

Step 2: Put all points on graph paper

We have three points (0, 4), (3/2,2)  and (3,0)

Now we will put points on graph paper.

Step 3: Now join the points.

How to Find Solution for System of Linear Equation:

System of linear equations means system will have more than one equation. To find solution for that we can apply both graphical and algebraical method. For graphical method we will draw garph for each equation,then there may me one of three condition

1. Lines are parallel: It means system have no solution.

2. Lines are overlapping on each other: It means system have no solution

3. Lines are intersecting at one point: It means system have one solution

One Step Equation Activities

One step equation activities contains a collection of problems that are one step equations. The one step equations activities problems that involve one variables and simplifying that equation to find the result. To solve the one step equation move the variables on one side and solving that equation. In this article let us see sample problems of solving one step equation activities.

One Step Equation Activities:

solving one step equations:

To solve a one-step equation, we must make the variable alone on one side of the equation. This can be done by using inverse operations. Addition and Subtraction (or adding the opposite) are inverse operations. Multiplication and Division are example of inverse operations.
Whatever you do to one side of an equation, you must do to the other side (like a balance scale).

Example1:

Simplify  `X/15` = 10

`X/15` ·15 = 10 · 15

X = 150

Example 2:

Simplify      x +3 = 8  , 3 is being added to the variable, so we must subtract 3 (or add -3) from (to) both sides

x = 5

Example 3:      

Simplify    9 = y – 4 , 4 is being subtracted from the variable, so we must add 4 to both sides

y =13

Example 4:

y - `1/6` = `1/12` , `1/6` is being subtracted from the variable, so we must add `1/6` to both sides

y = 3/12
Example 5:

Simplify         -30.2+ 5.1= n – 40.1

n = 15

first we must add -30.2 and 5.1, 40.1 is being subtracted from the variable, so we must add 40.1 to both sides

Example 6:

Simplify 6 – x =18 , because the 6 is positive, we must subtract 6 (or add -6) to both sides,

-x = 12  then the opposite of x is 12, so x is – 12

x = -12

One Step Equation Activities:

Example 7:

Simplify  X – 14 = 20

X – 14 + 14 = 20 +14,    14 is being added to the variable, so we must subtract 14 (or add -14) from (to) both sides

X – 14 + 14 = 20 +14

X = 34.

Example 8:

Simplify   4X = 12

`(4X) / 4` = `12 / 4` ,        `1/4` is being subtracted from the variable, so we must add `1/4` to both sides

`(4X) / 4 ` = 12 / 4

X = 3