Wednesday 30 November 2011

Linear/non-linear functions, equations, inequalities for Grade VII

Linear functions: 

Functions which have x as the input variable, where x has an exponent of only 1.

For example:
y = mx + c
Here we can see that x has an exponent of 1 in each equation.

The above functions such as this yields graphs that is straight line, and, thus, the name linear is alloted to it.

 

Linear functions are of three categories:

1) Slope-Intercept Form

2) Point-Slope Form

3) General Form


 

1) Slope-Intercept Form

A simple way to define a linear function is through the slope-intercept form of a linear function. When drawn on a common (x, y) graph it is given as:

y = mx + b

Or, in a formal function definition:

f(x) = mx + b


 

Basically, this function describes a set, or locus, of (x, y) points, and each of these points are located in a straight line. 'm' represents the slope of the line. 'b' represents the y-coordinate for the spot where the line crosses the y-axis and this point is called the 'y-intercept'.


 

2) Point-Slope Form

In the 'point-slope form' for the equation of a line, the definition does not start with 'y =' or 'f(x)=', so it's not written in a common function definition form.

But it can be very well written in algebra:

y = m(x - x1) + y1

and as a functional definition it can be written as:
f(x) = m(x - x1) + y1

To conclude the equation with the given slopes and points, we can say that the variable m is the slope of the line and the point x1, y1 are the points on the line. If anyone knows the slope of and the coordinates for one point on the line, then the he or she can include or can enter those values into this equation, and the equation would then define a set, or locus, of all the points on that line.


 

3) General Form

Let's say A, B, and C are three numbers forming an equation where the same equation can be assumed as:

0 = 2x + 3y + 4


 

All the points with (x, y) coordinates that can make the above statement true forms a line.


 

Using algebra, this general form can be changed into a slope-intercept form, and then you would know the slope and y-intercept for the line.

 

Non-linear Functions:

 

A non-linear function is defined as a polynomial function of two or even higher degree. The linear function is a polynomial function of degree 1. A quadratic function is a polynomial function of degree 2, defined by an equation of the form

y = ax2 + bx + c

The degree of a polynomial function is the degree of the polynomial itself.

For example:

Let us find the degree of the following function and also the points where x and y intercepts.

y= x2-2x-3

The degree of the given equation is 2. To find the point where its graph cuts the x axis, we can make y = 0. now we get,

x2-2x-3 = 0

Solving the equation we get x = 3 and x = -1. These are points where the function graph intercepts the x-axis.
To find the point where the graph intercepts the y-axis, assume x = 0. Now we get y = -3.

 


 

Linear Equations:

 

A linear equation appears like any other form of equation. A linear equation is made up of two expressions sets that are equal to each other. A linear equation is different as, it can have one or even two variables.

No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction. Linear equations graph as straight lines.

A linear equation in two variables shows an experiment between two variables in which the value of one of the variables depends on the value of the other variable. In a linear equation in x and y, x is called an independent variable and y depends on it. We call y the dependent variable. If the variables have other names, yet do have a dependent relationship. The independent variable is plotted along the horizontal axis. Most linear equations are functions (that is, for every value of x, there is only one corresponding value of y). When you assign a value to the independent variable, x, you can compute the value of the dependent variable, y. You can then plot the points named by each (x,y) pair on a coordinate grid.

 

Lines with the Same Slope

Lines with the same slope are either the same line, or parallel lines.

 

Consider 2x + y – 6 = 0. This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.

1. The actual equation is 2x + y – 6= 0

Subtract y from each side. 2x + y – y – 6 = 0 – y

2x – 6 = 0 – y

Multiply each side by 1.

 

1(2x – 6)= 1(y)

2x + 6 = y

 

  1. Show the original equation. 2x + y – 6 = 0

Add 6 to each side. 2x + y – 6 + 6= 0 + 6

2x + y = 6

Subtract 2x from each side. 2x – 2x + y = 6 – 2x

y = 6 – 2x

The given two equations, 2x + 6 = y and y = 6 – 2x are equal because you can change one equation into the other by using the symmetric property of equality, which states that if a = b, then b = a and the commutative property, which states that a + b = b + a.

commutative property 2x + 6 = y 6 – 2x = y

symmetric property 6 – 2x = y y = 6 – 2x

 

Nonlinear Equations:

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

For example:

7x5+ y = 0 and x3+ 13x2-4xy2 = 0 are the examples of nonlinear equations. We can also consider that the form x1/3 + y1/3 = 0 is a nonlinear equation.

Lets take an example of an equation:
(x+ 2)2= 6

This is a non-linear equation because, as per the definition, an algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Considering the equation (x+ 2)2 = 6

We can see that x can be concluded as x raised to the power 2 or x2.
Therefore, (x+ 2)2= 6 is not a linear equation.

Linear inequalities:

Phenomena of linear inequalities is a set of linear inequalities that anyone comes across all in the same time. Normally, a person starts with two or three linear inequalities.

 

For example:

  • Solve the following system:

    2x– 3y < 12
    x + 5y < 20
    x > 0

    It is better to solve a large number of linear inequalities for "y" on one side. Solving the first two inequalities, get the rearranged system as:

    y > ( 2/3 )x – 4
    y < ( – 1/5 )x + 4
    x > 0

Nonlinear Inequalities:

To solve non linear inequalities, We can use the following steps the following steps:

1.Place every component on the left hand side so that we have for example equation is greater than (>) 0.

2.Factorize and set equal to zero.

3.Solve and put the answers on the number line. Doing so it will divide the number line into two or three parts.

4.Select the value of each part and analyze them for each region and put that acquired value into each of the factors. Put '+' or '-' over the region looking into the values decide positive or negative values.

5.If the region has two '+' or two '-' then the region is positive. If the region has one of each then the region is negative.

6.If the inequality is "<" then include the negative regions.

7.If the inequality is ">" then include the positive regions.

8.If the inequality is a less than or greater than or equal to then include the endpoints with solid dot and the interval []

9.If the inequality is a less (greater) than then do not include the endpoints by showing an open dot and the interval () Lets take an example, Solve x2 + 3x > -2 1.x2 + 3x + 2 > 0 2.(x + 2)(x + 1) = 0 3.x = -2 or x = -

1 This divides the number line into three regions.

4.For the left region we choose-5 and have, -5 + 2 < 0, -5 + 1 < 0 5.For the middle region we choose -1.5, and get, -1.5 + 2 > 0, -1.5 + 1 < 0 For the left region we choose 0 and get, 0 + 2 >0, 0+ 1 >0 6.

The left and right regions are positive and the middle region is negative. 7.The inequality acquired is a greater in quantity we have the solution: (-,-2) U (-1,)

 

 

 

Monday 28 November 2011

Grade VII Percentage and Proportions

Hello my lovely students, once again your friend, your tutor is here to help you all in overcoming the mathematics problems. In this article, the main point of focus is on two math topics that is taught to students of class VII. Generally students of the age group of 13-14 study in this class. In this class students learn many important topics. Actually not only in this class but in all classes whatever you study is useful at some stage of life. You must have heard a common proverb “Knowledge is like a garden: if it is not cultivated, it cannot be harvested”. So kids start practicing each topic of mathematics and try to solve all the problems. If you work hard then definitely you will get the output, not only output but a positive output.

 

Now, let’s do some study of percentage of grade VII. What do you understand by the term percentage and from where this word is originated, any idea my dear friends? Yes you are right it is a Latin word that means for every hundred. If we bifurcate the term percentage as: per and cent then it simply gives its meaning i.e. per hundred. Learning percentage is also as simple as its definition. If you have to calculate the percentage then it must be like this

 

Amount

x100 = Percentage

Total

 

 

This formula can also be given as:

 

x

x100

y

 

It is used to express the value of x as percent of y.

 

Let us see, how to implement the values in the formula and how to calculate the percentage. Suppose we have an example given as: A person is having 500 stocks in a company, and the company is having a total share stock of 5000. How much of the company do that person owns.

 

For, the solution of this question, we will first determine what is the total amount and what’s percentage is to be calculated. In the above question total stock is company’s share that is 5000, and the person’s share is amount which is 500 stocks.

Now, put both the value in the formula:

 

Amount

x100 = Percentage

Total

 

 

500

x100

5000

 

In this we can simply cancel the zero’s so first cancel the 500 zero with the zero of 5000,

On doing this we get,

5

x100

50

 

Now, single one zero of 100 with the zero of 50, this will give us,

5

x10

5

Now, simply cut 5 from 5 on doing this only 10 remains,

This shows that the person is having 10% stock in the company. This is a simple example of percentage.

 

In percentage we also deal with increment and decrement. Finding the percentage of change is using the ratio of the amount of change to the original amount. The increased amount reflects the increased percentage and the decreased amount shows the decrement and sometimes it may be a negative value. So whenever you find percentage try to find that whether it is decreasing or increasing.

The Formula for increment and decrement is given as:

Increase % = 

Increase

x100

Original

In the same way decrease percentage is given as:

Decrease % = 

Decrease

x100

Original

 

To understand the concept of increment and decrement let’s see an example.

The amount of a pen is decreased from Rs 16 to Rs 14. Find the percentage decrease in the price of pen?

How you will solve this problem in this we can’t use the formula which is used to express the value of x as percent of y. In this we will go with the formula of decrease % as in question it is defined decreased price. 

First write what things are given to us:

Original price of pen = Rs 16.

Reduced price of pen is = Rs 14,

Now, decrement in price is= original price – reduced price,

                                          = Rs 16 – Rs 14,

                                           = Rs 2.

Now, Decreased % = 

Decrease

x100

Original

.

 

We have all the values to be put in the formula:

                             

2

x100

16 

=

1

x100

8

=

 

25

2

 

= 12.5 %.

Thus, the decrement in the price of pen is 0.5 %.

Hopefully you must have understood the question and with this the concept of decrement. In the same way we calculate the increased percentage. For this also have an example so that I can clear all your doubts and with this you can easily understand the base of the problem and how to solve. If you get the central idea of the problem then it will be very easy for you to solve all types of problems. But before having example I would like to tell you one thing that same formulas are used in case of population increment and decrement. So, here is an example to see the increased population percentage:

The population of a town is 2500 last year. This year it reached to 4000. Determine the total increase or decrease in the percentage of population of town.

Solution: first kids, find whether there is increase or decrease in percentage. In this question we can simply understand as the population was 2500 last year and now 4000 so there is increment in the population so percentage will also increase.

Given,

Population of last year = 2500.

Present population is = 4000,

Difference in population = 4000 – 2500,

Increased population is = 1500.

Put values in the formula:

Increase % =

Increase

x100

Original

=

1500

x100

2500

=

15

x100

25

=

3

x100

5

=  3 X 20 %,

= 60%.

Thus the increased % of population is 60%.

In this way we can deal with all types of percentage questions without any problem.  You can also switch to online service providers that help you in solving all the problems related to all math topics. Thousands of websites are playing there games to help you and to solve your math problems easily and more effectively so that, kids can learn all the topics easily and score good grades.

 

Now, we will matriculate proportional relationships. What do you mean by Proportion?  Proportion is a name given to a statement that two ratios are equal. I think you all must know of ratios as you all read this in the pervious classes still I want to provide a basic idea of this.  Ratios are defined as a quantity used to compare two numbers. In ratios we separate the two numbers with a colon (:) and in proportion we use double colon (::). For illustration, if you have to write the ratio of 4 and 12,

Then we will write it as:

4:12 or it can also be represented as a fraction like 4/12 and in mathematical language we will call it the ratio of four is twelve.

Proportion defines the ratio of two numbers to it expresses their proportional relationships.

Like,

A/B = C/D in other terms it can also be represented in a way in which we use colon, like

A:B = C:D,

Or,

A/B::C/D

If the two ratios are equal then the cross products of the ratios must be equal, if not then ratios are not equal.

For proportions,  

 

A:B = C: D,

A X D = B X C.

 

Now, have an example of proportions to understand it more properly,

48/21=81/x ,

In this proportional relationship we have to calculate the value of the x variable,

You all must be familiar with the cross multiplication; in cross multiplication we take the right side denominator value to other side and on numerator and vice-versa,

 

So, take the x to left side and multiply it with 41 and 21 on right side and multiply it with 81.

See how,

X  * 48 = 81 * 21,

48x = 1701,

X = 1701/48,

X = 35.4375,

And we can write it as 35. 4. Remember the rounding off values, which you have learned in pervious class, in which we write the values after decimal in round figure.

 

Now, let’s see one more example, in which we see both the percentage and ratio. As I told you that all the math topics are interrelated so, there is an example to see how both topics that we studied today are related with each other.

Question: A mixture of milk and water, are mixed in the ratio as4:1. Find the percentage of milk in the mixture?

Solution:  in the above question, Milk: water = 4:1 so,

 

% of milk is =4/4+1,

 

 = 4/5

 =  

4

x100

5

= 80 %.

In this way you see how we are solving problems in percentage.

Students, this is all about both the topics from my side. If you still have any query related to this topic then you can take help of online math service providers. These online math service providers help you in learning different math topics and in solving the problems related to math. In this online help, you will find several kinds of solvers which directly solve your entire math problem and help you in overcoming them. Now, onwards if any problem occur switch to online help rather then wasting many hours on single problem. So, take online help and say bye to math problems.