Lines in Grade VII

Hello friends in this article we are going to discuss important geometrical topic that you study in grade VII. A line can be defined as a continuous extent of length, straight or curved, without breadth or thickness; The notion of the line was very first introduced by ancient mathematicians to represent straight objects and things with negligible width and depth. in more mathematical term the same can be defined as a geometrical object that is straight, infinitely long and infinitely thin. There are different types of lines like straight line, parallel line, perpendicular lines, intersecting lines, many more. Properties mean any qualities, features that are embedded in line.

Let’s focus on the properties of parallel lines. Parallel lines are those lines which move right next to each other and they move infinitely long but never cut each other. Properties of same said that, the two lines lie in the same plane, and do not intersect or cross each other. The properties of parallel line based on the Euclid’s parallel property means they can’t cut each other at any point. In other word the property says that lines are nothing with a pair of lines in a same plane which do not cut or meet each other. In this, we can introduce the other line called as transversal line that crosses a pair of parallel lines on a slant.

 

Let’s have an example of the properties of the parallel lines.    

Example 1:

Define the equation parallel to line 4q + 4p = 8 with the point (-6, 4).

Answer:

Given,

 4q + 4p = 8 and the point (-6, 4)

To detect the parallel line, we have to find the slope first.

For finding the slope, we need to change the given equation into slope intercept form.

4q + 4p = 8

Do addition of 4p on both sides of the equation,

4q + 4p = 8

- 4p = -4p

4q = -4p + 8

Divide by 4 on both sides,

q = (-p + 2)

The obtained equation is in the form, q= mp + b

So, the slope from the obtained equation m = -1

Generally, we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = -1

So, m2 = -1

The line equation is,

(q - q1) = m(p - p1)

(q - (4)) = -1 (p - (-6))

(q - (4)) = -(p + 6)

q - 4 = - x - 6

Subtract 4 on both sides,

q = -p - 2

Output: Thus, the pictures of parallel lines is given through the lines q= -p - 2, 4q + 4p = 8.

In this way we solve problems related to parallel lines. Geometry is very interesting mathematical part, that you can easily understand with help of bit practices. So do practice and score good marks and feel proud.

Angles in Grade VII

Students, today I will make you aware with geometry of Grade VII. Before we move to any topic or any other things tell me what is geometry? Geometry is any well defined shape; each solid object has its own geometry that describes its shapes, properties, etc. Geometry is all about shapes, if you get familiar with shapes then you are the hero of geometry. as you study algebra from primary grades similarly you will also learn geometry from primary schools. As you grow up and grade changes complexity of topics also changes.

Today we are going to talk about Complementary and supplementary angles. First go through complementary angle. Two angles are called as complementary if the sum of their angles is equal to 90⁰. If one angle is known, than its complementary angle can easily be determined by subtracting the measure of its angle from 90⁰.   

For instance: What is the complementary angle of 47^o? 
Solution: 90^o  -  47^o  =  43^o

Whenever we have to determine the complementary angle and one is given minus it from 90 and the resultant angle is determined in this way.

Example 2: if we have two angles as 27⁰ and 63⁰, are they complementary angles or not.

Answer: we have two angles,

One is 27, and other is 63,

If we add both the angles then they will give result as: 27 + 63 = 90,

Hence it is a complementary angle, as sum of both the angles is equal to 90 which is the condition of complementary angle.  

 

Now, the other type of angle i.e. called as the supplementary angle, if the sum of two angles is equal to 180 then such kind of angles are named as Supplementary angles. When one angle is unknown to you and you have to determine the unknown one then you can easily calculate it by subtracting the given angle from 180⁰.

Let’s see how to calculate the unknown angle with help of example.

Determine the second supplementary angle when one is 143^o? 
Answer: 180^o  -  143^o  =  37^o

Now the other example of the same, calculate that the given angles form a supplementary angle or not?

 Suppose you have two angles, 100⁰ and 80⁰,

You can easily say that both angles results, supplementary angle. Because when we add them they will give result as:

100⁰ + 80⁰ = 180⁰

Thus, satisfying the condition of the supplementary angle.

Example: one angle is 24⁰ and other is 58⁰  tell whether it is complementary or supplementary angle.

Solution: Angle one = 24⁰, second angle = 58⁰,

first + second = 24⁰ + 58⁰,

                            = 82⁰.

As on adding both angles we are getting only 82, so it is neither a complementary angle nor supplementary. Because for complementary angle it must be equal to 90⁰ and for supplementary it must be 180⁰.
In the upcoming article we will talk about other topics of geometry and if you have any doubt than you can take my help easily. I am here to help you and to solve your mathematical problems. 

Multistep Problems in Grade VII

Algebra! Who invented this boring branch of mathematics. Many students usually ask this question. But my dear students I am here to take your pain and to make this boring subject an interesting one for you. Whenever you deal with any math problem try to understand the problem behind the concepts and relate it with the real life problems then surely friends you will understand the problem properly. Now, move towards the today’s topic that is multistep problems in grade VII. What do you understand by the term multistep problems? As the name says multistep, you easily understand its concept that multistep means multiple step and by multistep problems we mean that how to solve problems involving different steps.

In multistep problems, a problem is solved in different steps and each and every step is very important. If you solve any problem and if any of its step is wrong then you won’t get the final answer right. In such type of problems each and every step must be right otherwise the output will be wrong even a minute mistake can cause problem to your answer. so be careful whenever you solve these types of problems and check and all the steps properly and carefully. The multistep problems can be in any form like they can be in equation form or word problem form.

Let’s take an example and understand the problem properly.  

First take an example of equation which is solved in multistep.

Example: Solve the given equation for the variable q

20 = 3(2 q + 8) + 4

Answer:

-20 = 3 (2q + 8) + 4

Eliminate the parentheses,

-20 = 3 (2 q) + 3 (8) + 4

-20 = 6q + 24 + 4

-20 = 6q + 28

Subtract 28 on both sides of the equation

-20 - 28 = 6q + 28 – 28

-48 = 6q

Divide by 6 on both sides of the equation

-48/6 = 6q/6,

-8 = q

This is the final answer of the equation.

 

 

Now, here is an example of the same but the form of word problems.

Question: Allen earns a base salary of $94 per week with a commission of 14% of sales. if she had $100 in sales last week, then find her total pay.

Solution: as you all know the formula to calculate commission and total pay.

Commission = commission percentage x sales,

Total pay = base salary + commission.

First calculate the commission. For this we can use the first formula

Commission = commission percentage x sales,

= 14 % x 100

= 14 /100 x 100,

= 14.

so the commission is $14.

Now we can easily calculate the total pay, using second formula,

total pay = base salary + commission,

= 94 + 14,

= 108,

The total pay was $108.  

in this the simple way to solve multistep problems only take care that all the calculations and formula used are right or not, then you can easily solve all the different problems.

Linear Equations VII Grade

Hello my dear students, today we will focus on few more topics of algebra that you study in class VII. As you all now algebra is an interesting subject if you study it with interest and one of the boring subject if things go out of your mind then it is one of the difficult subject. In this article we will talk about Linear/non-linear functions, equations, inequalities. What do you understand by linear function? A linear function can be defined as a polynomial function of single degree. In this we relate a dependent variable with an independent variable in a simple way. It is that mathematical equation in which there is no independent variable that is raised to power greater than one. We can define the simple linear function as a linear equation which is having a single independent variable (y = a + bx), which results in a straight line when we plot the same on the graph.  

Now, let’s switch gear towards the form of linear function. In order to define the linear function we need an equation of first degree. The general form of linear function of first degree is given as:

f= (x, y)/ y = mx + b

In this m and b are constants, x and y are the linear function. When we plot the graph of same it will give a straight line. There are three main forms or types in linear functions.

1. Slope- intercept form: its general form is given as: y = mx + b,

 

2) Point slope for, its general form is given by m = (y –y1) / (x – x1)

3) General Form is given as: Ax + By  + C = 0.

Now, have a look on how to solve the linear functions. Solving the given function by altering its position and simplifying the equation to get Y is generally known as solving linear equations. The solution of these equations provides solutions of the corresponding practical problems. Here are few steps that you can use when you deal with the linear function in word problem form:

1. Read the given problem properly and then note down what is given to us in problem and what other things are required.
2. Denote the unknown quantities by literals like x, y, z, u, v, etc.
3. Translate the statements of the problem step by step in mathematical form so that we can get what to do.
4. After that, look for the quantities that are equal. And then make the equations corresponding to these equality relations.
5. Use an appropriate method and solve the equation formed in step 4.
6. It is the last step and in this you have to check the solution of the problem by substituting the value of the unknown found in step 5. Now, let’s have few examples of linear functions so that you can easily understand the concept and process of solving linear function problems.  

Example 1: Solve for a and b, where, b = a+ 13 and 2a ? b ? 10 = 0

Solution: in this we can easily put the value of the b as b = a + 13 in the equation

      2a ? b ? 10 = 0  

 2a ? (a + 13) ? 10 = 0,

a ? 23 = 0,

a = 23.

Now put the value of a  23 in 2a ? b ? 10 = ,

? y + 36 = 0,

? y = ? 36,

y = 36.

After linear function it’s the time to move towards the next topic that is called as the non- linear function. In simple terms we can define non-linear function as an equation whose graph is not linear and an equation with degree two or more than two. The general form of the same is given as:

f(x) = +….+ a₁x¹ + a₀ where a₀, a₁ ..a_n are stables. In the non linear function a_n is defined as a primary co-efficient and a_n x_n is principal term. The greatest degree of non-linear function is greater than two or similar to two. In non linear function, graph can be a curve, zig- zag line, or any even shape but it can’t be a straight line. The Non linear function can be defined in quadratic form, exponential form and logarithmic form.

Different types of non linear function deals with solving several types of polynomial function where as polynomial function is also called as non linear function.  We can solve the non linear function using substitution method or quadratic equation operations, etc. 

 

Non linear functions play an important role in algebra. Any of the function is not a linear function and can’t be a complete linear function by transforming the Y variable.  

Normally, there are three types of non linear function that we use in mathematics, they are given as:

1. Exponential function
2. Quadratic function
3. Logarithmic function

This is the time to see few examples of the non linear functions.

Example 1: x² - x – 12 = 0, calculate the value for ‘x’ for the given non linear function.

Answer:

As you all know the given equation is in form of x² – 4x +3x - 12 = 0, which resembles with the general form of quadratic equation.

Now, we can get the value for x from the primary term and 3 from secondary term.

x (x – 4) + 3(x - 4) = 0

Now combine the like term (x -4)

(x + 3) (x - 4) = 0

To get the value for x we can associate the factor to zero

a + 3 = 0 or a – 4 = 0

a = -3 or a = 4

a = 4

Thus, the factors a₁ and a₂ are -3, 4.

Example 2:

p² - 3p – 10 = 0, determine the value for p for the non linear function.

Answer:

Now, we can find the factor for the given quadratic equation

p² – 5p +2p - 10 = 0

Now, get the value for x from the primary term and 3 from secondary term.

p(p – 5) + 2(p - 5) = 0

Now we can combine the similar term (p - 5)

(p + 2) (p - 5) = 0

To get the value for x we can associate the factor to zero

p + 2 = 0 or p – 5 = 0

p = -2 or p = 5

Thus, the factors p₁ and p₂ are -2, 5.

Example 3:

x² - 6x + 5 = 0, get  the value of x for the given non linear functions

Solution:

First we can learn the factors for the given quadratic equation

Sum of the roots (-6) = (-5) + (– 1)

Product of roots (5) = (-5) × (-1)

By combining these two roots we can obtain the factor form as,

x² – 5x - x + 5 = 0

Now get the value for x from the primary term and 5 from secondary term.

x (x – 5) - 1(x - 5) = 0

Now we can combine the similar term (x - 5)

(x - 5) (x - 1) = 0

To get the value for x we can associate the factor to zero

x - 5 = 0 or x – 1 = 0

x = 5 or x = 1

Thus, the factors are x1 and x2 and the values are 5, 1.

Now, switch to other topics Equations and inequalities. You all are familiar with both these terms. Here, I will give a brief introduction of both the topics as you are reading the same from earlier classes.  Equations are those mathematical statements that are joined together with help of equal to symbol. There are many different types of equations like linear equations, quadratic equations, polynomial equations, and many other types of equations. There are several different types of methods that you can use to solve different types of equations like linear equations can be solved by simple motion of the variables and constants.

Inequalities can be defined as an expression which is defined using special symbols.  Mainly four different inequality symbols are used using which inequalities are defined like: greater than (>), greater than equal to (<), less than (>=), less than equal to (<=). Solving linear equalities is very simple and you can easily solve them. Solving linear equalities are very similar to solving a linear equation. Inequalities give infinite number of solutions and the answer is right only when the inequality is true.

In all the above mentioned topics for any problem you can take online help and solve all your problems.       

Numeric Expressions in Grade VII

Mathematics is a wonderful language that deals with different types of numbers. Math is a broad subject that is divided in several different types of branches. In this section we are going to focus on the few important topics of algebra a pure branch of mathematics of grade VII. In this grade students usually study algebra 2. Algebra is nothing but it is simply the movement of the numbers, variables, and constants. If you love playing with numbers then definitely you we will love this subject. Now, come to the topic that is simplifying numerical expression.

Before we move to simplification of the numerical expression, let’s talk about the expression and simplification individually. Expression is a finite combination of the symbols that is well-defined and formed according to rules that depend upon the context. Symbols can designate numbers constant, variables, operators, functions, and other different types of mathematical symbols. These symbols includes punctuation, symbols of grouping, and other syntactic symbols. In algebra we generally deal with two different types of expression one is numeric and other is algebraic expression. An algebraic expression is that mathematical phrase that contains ordinary numbers, variables such as a, b ,x, y,.. and operators like add, subtract, multiply, and division. Let’s have a look on different types of algebraic expression.

a+ 2,

x- y,

6x,

c- 3/4d and other such type of expressions.

Now, the other type of expression that is called as the numeric expression, this means involving numbers and the term expression means “phrase”. In simple terms we can describe the same as: a mathematical phrase involving only numbers and one or more operational symbols.
Here are few examples of numerical expression,

5 + 20 – 7,

(1 + 3) – 7,

 (6 × 2) ÷ 20,

 4 ÷ (20 × 3),

7 × (4² + 3)

 

A numerical expression is the group of numerical or numeral values, which are separated by addition or subtraction. Numerical expression is simply the real number or all the positive values(1,2,3,4,5,6…) and negative values(-1,-2,-3,-4,-5,…) and zero. Order of process is one of the methods used to evaluate the numerical term in given expression.

Now, talk about simplification, the term simplification means reducing any expression to the least value for where we can divide the expression again. In other words we can say that simplification means solving the expression and removing all the complex or simple operators from the expression.  Simplification of numerical expression means removing all the possible operators from the expression and taking the same to the simplest from, which will not contain any type of operator. The different numerical expression will always contain the numeric values only; They never contain the variables or any alphabetic value. While on the other side algebraic expressions contains both the numeric as well as variables with different type of operators.

Simplification of Numeric expression is very simply, if you knows how to perform different operations like addition, subtraction, multiplication, and divisions. All the operations are very simple and you have learned how to solve the problems of such type of operators in the early classes. This is the time when you have to implement that properly and solve different types of problems. In this article I will teach you how to simplify numeric expressions you can easily solve them by performing simple operations. Order of process is used when one or more function is included in any problem. We use three rules in order to deal with such kind of problems in order of operations.

> First one is perform the action within the parentheses.  

> From left to right order, do all the multiplication and division function.

> Finally do all the addition and subtraction operations from left to right.

Let’s have few examples of the numerical expression and see how to apply above mentioned rules on the numerical expression.

Example: Simplify the numerical term in the expression 11+ 7 x (6 + 3) ÷ 9 - 8 using the order of operations.

Solution:

In this problem all the four symbols are present; we have to do all the four operations that includes: additions, subtraction, multiplication, division operations here.

= 11 + 7 x (6 +3) ÷ 9 – 8

=11+ 7 x 9 ÷ 9 – 8

= 11+ 63÷ 9 – 8

= 11 + 7 – 8

=18 – 8

= 10

Answer: 10

In this way we simplify the numerical expression. In the above expression we first solve the parentheses then on the priority basis we first perform the multiplication operation, then division after that addition and in last subtraction operation.

Example 2:

Solve the numerical expression 100÷ (13 + 3 x 9) - 2 using the order of operations.

Solution:

In this expression we are having all the four basic mathematical operators. So we have to perform all the four operations.

= 100 ÷ (13 + 3 x 9) – 2

=100 ÷ (13 + 27) – 2

= 100 ÷ 40 – 2

= 2.5– 2

= 0.5

Answer: 0.5

Example 3:

Get the answer of the given numerical expression 55 - (8 * 9 - 7) + 2 using the order of operations.

Solution: In the above mentioned expression we are having three operators and according to rule we first solve the parentheses then multiplication, after that addition and finally subtraction.

= 55 - (8 * 9 - 7) + 2

= 55 - (72 - 7) + 2

= 55 - 65 + 2

= 55 - 67

= -8

Answer: -8

Example 4:

Evaluate the numerical term in the expression 3 + 5 (2 + 4) - 7 using the order of operations.

Solution:

Since all the four symbols are present in the given expression, we have to do all the three additions, subtraction, multiplication, division operations here.

=5 + 5 ( 2 + 4 ) - 7

= 5+10+20 -7

= 35-7

= 28

Answer: 28

In this way we deal with the different type of numerical expression and simplify them to the simplest form. Whenever you deal with simplification of numerical expression always remember the order of operation rule and solve the problem using the same rule. If you don’t use the order of operation rule than the simple problem become the complex one and difficult to solve.

Now, talk about the other important topic that you study in grade VII i.e monomials.   

A monomial is that term which is comprised of a combination of the following: numbers, variables, and exponents. In algebraic expression and equations, terms, or monomials are separated by addition (+) and subtraction (-) symbols. In simple terms we can define a Monomial as an algebraic expression with only one term. For instance, 7ab, – 5p, 3z2, 4po, 5 etc. Monomial may have a constant or variables or both. To denote variables we use letters a,b,c ,d, x, y, l, m, ... etc. A variable can take various values; its value is not fixed. On the other side. a constant has a fixed value such as: 4, 500, 678,12432535, 100, –17, etc. Let’s put some light on the examples of Monomials

1. 19xy
in this, Coefficient: 19 , variables x and y and exponent 1

2. -2ab²
in this expression  Coefficient: -2 , variables are ‘a’ and ‘b’ and exponent is 2

3. 41 ab³
Coefficient: 41 , variables are a and b and exponent is 3

4. xa²
Coefficient: -1 because -x² is the same as -1x²

Variable is x and the exponent is 2.

Let’s switch to the point that what operations can we perform on Monomials,

1.    Addition and Subtraction of Monomials: When terms or monomials contain the same variable and same exponent, they are similar terms.

Addition and subtraction of monomials is done by combining the like terms. In addition we add the similar terms and in subtracting we subtract the similar or like terms.

Simplify the following expressions.

1)      7 + 7x +13x

In this simply add the x terms and leave the other terms as it is,

 20x + 7.

2)      -12c + 12c,

In this we are having both terms as the co-efficient of the c one co-efficient is -12 and other is 12. On simplifying the term we get 0.

3)      8y - 3y

Both the terms are of y on adding we get 5y.

4)      x² + y² + x

we can’t simplify the above problem further as there is no like present in the Monomial.

Multiplication of Monomials: When you multiply the monomial, first step is multiplying the numerical coefficients (for e.g. 4 and the 8) and then multiplying the literal coefficients or variables (a and b). in the second step you need  to multiply the similar variables by adding their exponents (for e.g. 3+2). (Rule am * an = am+n).

Simplify the following monomials:

1) 5 ab * 5 b = 25 ab²

2) 2 xy * 3 yz = 6 xy2z

3) -4 x6 * 6x2 =  -24 x8

4) 4 b5c * 7 ab2c = 28 ab7c2

5) 20 ac * pq = 20acpq

Division of Monomials: When we perform this operation of the monomial the very first step is dividing the numerical coefficients (for e.g. 24 and 8) and then dividing the literal coefficients or variables (a and b). Second step is to divide the like variables by subtracting their exponents (for e. g. 5-2 ). (Rule am / an = am-n).

Simplify the following monomials:

1) 25 pq / 5 q = 5 p

2) 15 ab⁴c / 3 bc = 5 ab³

4) 49 x²y⁵z / 7 xy²z = 7 xy³

5) 20 ab / ab = 20

 

 

 

 

Graphing Data in Grade VII

Hello friends! Today we will discuss two interesting topics of algebra that you study in grade VII. The first topic that we will cover today is Graphing data to demonstrate relationships and the next will Arithmetic sequence. Graphing is a simple pictorial representation of any data on the number line. With the help of graph you can easily understand and easily analyze the data.  Data recorded in experiments or surveys is exhibited by a statistical data graph. We use different variant types of graphs for the representation of the statistical data graph. Mainly, we have eleven types of graphs that we generally use for the representation of the statistical data.

1. Box Plot: It is a convenient way of graphical representation of data or we can say it is a way of summarizing a set of data measured on an interval scale. It is sometimes used for the exploratory data analysis.  In box plot graph we show the shape of the distribution, its central value, and variability.
2. Stem and leaf plot: this is a method used for showing the frequency with which certain classes of values occur. In this we can make a frequency distribution table or a histogram for the values, or we can easily use a stem-and-leaf plot graph method.
3. Frequency polygon: it is a graphical display of a frequency table, in this intervals are shown on the X- axis and the number of scores in each interval is represented by the height of a point situated above the middle interval. The points are connected in such a way that together with the X-axis they form a polygon.

 

1. Scatter plot: it is a graph in which we use Cartesian coordinates to display values for two variables for a set of data. The data is represented as a collection of pints, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis.   
2. Line graph: this type of graph displaying data or information that changes continuously over time.

 

 

1. Bar graph: A bar chart or bar graph is a graph with rectangular bars with lengths proportional to the different values that they represent. The bar graph can be plotted on horizontal axis or on vertical axis.

1. Histogram: it is a graphical representation of data that shows a visual impression of the distribution of data.

 

1. Pictograph: They are often used for the representation of the data in graphical form.
2. Map graph: This type of graph is also called as the grid system. in this is the standard grid of up and down drawn lines and left and right drawn lines creating a grid of intersecting lines.

 

1. Pie graph: The pie graph is the most commonly used statistical charts for the representation of the business data and the mass media data.

 

1. Line plot: it is a graph that shows frequency of data along a number line. It is one of the best way for the representation of the data.

 

 

 

In Grade VII, students learn histo graphical representation of the data, frequency polygon and the bar graphs. In Statistical data graph, a histogram is used for the representation of the two dimensional data used to graph continuous data. Histograms do not have spaces between the bars.

Here is an example in which you see how to represent data in form of histogram graph.

 

.

Data	Frequency
0-10	5600
10-20	5250
20-30	4750
30-40	3750
40-50	3500
50-60	2750
60-70	2600
70-80	2750
80-90	2500
90-100	2400

Next method is frequency polygon, in this we use to study the graphical representation of a frequency table. This type of graph offers approximate smooth bend that explains a frequency distribution if the class intervals were to be made as small as possible.

The study of points is involved so that collectively with the X-axis they form a polygon.

Statistical data graph in Frequency polygon problem:

Construct the Statistical data graph using Frequency polygon.

Data	Mid term value	Frequency
0-2	1	1
2-4	3	12
4-6	5	9
6-8	7	1
8-10	9	1

Graph:

 

Data Graphs Using Bar Graph

Other graphs that you study in the same class is called as bar graph as I already mentioned earlier in the article that what it is? Let’s take an example of this, and see how to plot such kind of graph.  These graphs are one of the most interesting topic that you can easily plot.

 

Here is an example of construction of bar graph for the statistical data graph.

Brain region	Receptor Binding
1	35
2	50
3	28
4	20

Graph:

 

This is all about the different types of graph that you use for the demonstration of the data. Now, we will switch to the next topic that is Arithmetic sequence. It is also known as the arithmetic progression, or arithmetic series. An arithmetic sequence consists of the sequence of numbers and accept the first term,  remaining terms can be obtained by adding one number to its preceding number. It is denoted as the arrangement of two consecutive numbers, the progression which is constant.  Other than AP we also have other type of series called as geometric progression, harmonic progression. You will learn all these in the future grades.

 

 

Arithmetic progression also have its own general form, if you see any such type of equation then understand that it is in form of A.P. The general form of an A.P. is given as:

 

a = first term, d = common difference, then A.P. is a, a+d, a+2d, a+3d,.....

In this every time you will observe that in any term the coefficient of d is always less by one than the number of terms in the series.

Thus, second term is a+d

third term is a+2d

fourth term is a+3d

tenth term is a+9d

and generally, n^th term is a + (n-1)d.

If n is the number of terms and if t_n is the n^th term, then t_n = a+(n-1)d.

 

 

In arithmetic progression common difference is calculated by subtracting any term from the series from the intermediate succeeding them.  Let’s see few examples of the arithmetic series.  

See the following series and find whether it is a A.P. or not?

i) 1, 3, 5, 7, …

In this example, common difference in the first sequence is 2 and each term in the first sequence is succeeding with the increment of the two and thus forming an arithmetic sequence.

ii) 3, 7, 11, 15, …

In this there is difference of 4, in all the terms present in the series. You can easily found the nth term by adding 4 in the n-1th term.

iii) 15, 12, 9, …

This is also in form of arithmetic progression; in this each term is decreasing by 3.

iv) x, x - d, x - 2d, .....

in the fourth it is -d

v) p , p + q, p + 2q, p + 3q,..

In this each time there is increment of q. so the series is in form of A.P.

We can also calculate the sum of the Arithmetic progression. The sum is some sort similar to normal addition, let’s have a look on how to do addition of the arithmetic sequences.

Let a=first term, d=common difference, l=t_n=last term, s=required sum. Then,

S= a + (a +d) + (a + 2d) +(a + 3d)+…..a + (n -2)d + a+(n-1)d

If we write the same series in the reverse order then we will get the sequence as:

 

S= a +(n- 1)d+ a+(n - 2)d+….+ a + d +a

Adding together the two series we will get the answer as:

2s = [2a +(n-1)d] + [2a + (n-1)d] +………up to nth term.

2s = n2a + (n -1),

S = n/2 2a + (n -1)d,

S = n/2a + a+(n-1)d

 

S = n/2(a +1),

Here, l = a + (n -1)d

 

Now have a look on the properties of the Arithmetic progression.

 

If p, q, r, s are in A.P., then

1: p +- k, q +- k, r +- k, s +- k,… are also in arithmetic progression.

2: kp, kq, kr, ks…. will also be in A.P.

 

 

A remark on finding a few members of an A.P. whose sum is given along with other conditions:

1. If the sum of three numbers in A.P. is given, take the numbers as a-d, a, a+d.
2. For the five number series always take take them as: a – 2d, a –d, a,  a+d, a +2d.
3. If you have to take a four numbers series then always take them as: a- 3d, a – d, a +d, a + 3d.
4. For any six term, take the series as: a – 5d, a – 3d, a – d, a+ d, a +3d, a +5d.

This is all about arithmetic progression. Using the above mentioned form, series, formula and properties you can easily solve different type of arithmetic progression problems.

Systems of Equations in X Grade

Hello my dear friends today we will learn few interesting topic of grade X that are imbibe in Algebra. Algebra is a very large subject that has lots of topics and you can easily learn them with intuition. Mathematics is that subject that can only be learnt if you have intrusion and is considered as one of the most interesting and important subject. Many students are math phobic and they always try to move away from this subject. Friends no need to get afraid of this wonderful subject you can easily learn this subject with help of you teachers, tutors and me. In this article we will focus on changing parameters of function, and system of equations. Before this in the previous sections I made you familiar with other math topics like, Different forms of numbers, Scientific notation, square roots, exponents, radicals, absolute value, factorial, logarithms., Properties of numbers, Estimation of solutions, Sequences and series Proportionality/direct variation, and exponential/fractional expressions.

 

Now, start talking about our today’s topic i.e. changing parameters of function. First talk about the parameters, in common meaning, the parameters are used to identify a characteristic, a feature, a measurable factor, which can be used to define a particular or specific system. For the evaluation or for the comprehension of an event, or a project or any other such type of situation, parameter is an important element. If we talk about parameters in Mathematics then a parameter is a defined as a quantity that serves to relate different functions and different variables using a common variable when such a relationship would be difficult to explicate with an equation.

 

Different Mathematical functions have one or more arguments that are designated in the definition by variables, on the other side their definition also have parameters. In the list of arguments variables are mentioned that the functions takes, but the parameters are not mentioned. When we define parameters or any parameters present, the definition actually defines a whole family of functions, one for every valid set of parameters value. We can take the example of the general quadratic equation form: that defines the function and parameters.

F(x) = ax² +bx +c,

In this equation, variable x designates the function argument, but a,b, and c are parameters that determine which quadratic function one is considering. The Parameters could be incorporated into the function name to indicate its dependence on the parameter. To understand this we can take example of log, such as: 

log_a(x) = log (x)/ log (a),

In this ‘a’ is a parameter that is indicating which logarithmic function is being used. Here ‘a’ is not an argument of the function.

 

Parameters varies from function to function, in Grade X, you commonly use population parameter. A quantity or statistical measure that for a given population which is fixed and that is used as the value for a variable in some of the general distribution or frequency function to make it is a descriptive of that population.The variance and the Mean of a population are referred as the population parameter.

The next topic that we will discuss today is called as system of equations. A system of equation means a set or a group of equations and sometimes we refer this as simultaneous equations also. In this we have more than one equation with multiple variables and we try to solve the equation in order to calculate the value of the all variables. Linear equations are very simple if we compare them with non-linear equation and the simplest linear system is one with two equations and three unknowns or variables. In simple words we can define the same as a collection of two or more equations with a same set of unknowns. For the solution of system of equations, we need to find values for every of the unknowns that will declare every equation in the system. When you deal with system of equations you can either have linear equation or non-linear.

 

Now, let’s focus on the types of systems of equations. In grade X you will learn Elimination, Substitution, linear equation, matrix, and consistent equations. Now, have an introduction of all the topics that I mentioned here. First talk about the elimination method or elimination technique, it is considered as one of the algebraic method that we use for solving the system of equations. In elimination method we perform operation on the one equation the operation may be of adding some number or multiplication, etc. we perform this operation in order to cancel one of the variable and with this we can easily find the value of other variables.

The next one is Substitution method, this method the algebraic expression of one of the variable is substituted in another equation at the place of the respective variable and then the variable is used to solve, when we solve the variable we get the numeric value and then easily with its help we can solve the different problems. by substituting in any of the equation the value of the second variable is also found easily.  The third one is Linear equation, this is also considered as an algebraic expression, which relates the two different types of variables and produces a graph which is always in the form of a line.

Fourth one is Matrix, it is a rectangular array of numbers and in this we use to write numbers inside the bracket. Matrix method is used to find the solutions for complex systems of equations. The last one is called as the consistent System.  In this we have set of equations whole solution set is represented by only one ordered pair. Here are few examples for you students so that you can easily understand how to solve the system of equation. Examples are the best way to understand different type of problems and with this you can solve multiple problems in an easy way.

 

Here, is the very first and simple example, give the solution of the problems and tell whether it is example of system of equation or not.

y + 21 = 71

Solution:

This is a simple example of linear equation as only one equation is given but in system of equation we have more than one equation that either is two, three or more. We need not to implement any of the above mentioned method in this as we can easily solve the problem by subtracting 21 from both the side of the equation.

 Y + 21 – 21 = 71 – 21,

Y =50,

In this way we can easily calculate the value of the unknown in this equation.

Now, here is another example for you. Give the solution of the problems and tell it is system of equation or not.

Example: b = 2a + 1, 2b = 3a – 2

Solution: this is a system of equation as it is having two equations, and for solving this problem you can easily use any of the above mentioned method which we use to solve the system of equations. Let’s use the substitution method and find the solution of this problem.

Step 1: substitute the one equation to another equation like:

2(2a + 1) = 3a – 2,

Step 2: now, we have single variable equation, we know how to solve that variable equation as the equation is now in form of single variable which you can easily solve and find its value.

4a + 2 = 3a – 2,

4a – 3a = -2 – 2,

a = -4,

In this way we find the value of the one variable now you can easily find the value of the other variable by putting the value of ‘a’.

Step 3: to find the value of b  put value of a

b = 2a + 1,

b = 2(-4) + 1,

b = -8 + 1,

b = -7,

Thus we find the value of ‘a’ and ‘b’ using substitution method.

One more example of the system of equations and solve this problem using the elimination method.

Example: 2p + 2q = 4

          4p – 2q = 8.

Solution: As there is two equation given to us, so we can easily say that it is a system of equation having two variables p and q.

Step 1:

2p + 2q = 4.

4p – 2q = 8.

Step 2: in this subtract equation 2 from equation 1 and on doing this we get,

           6p = 12

Step 3: now, divide both the side by 6, on doing this we get

p = 2.

 

Step 4: now, replace p with its value in the equation 1, when we do this we easily calculate the value of the other variable q.

 

 

 

4p – 2q = 8.

4(2) – 2q = 8.

8 – 2q = 8.

q= 0.

Now, we have both the value of p and q

(p, q) = (2. 0).

In other method you will study in the higher classes.

 

Keywords: