Variables in VIII Grade

Hello friends in today's session we are going to learn about variables, expressions, equations and inequalities. Let's just take them one by one. We will start with variables.

A variable in mathematics is a quantity which may change itself according to the given conditions. It is generally represented by English alphabets x, y z, a, b, c etc. If the given condition has no variables than it is said to have constants.

A constant is a value which remains unchanged like numbers 3, 4, 5 etc.

For Example- x + 2 = 6, where x is variable and 2,6 are constants.

Variables are further categorized as dependent and independent variables.

The variables whose value depend on other terms in the condition are known as the dependent variables whereas the ones which take different values freely are known as the independent variables.

When variables come together with some constants then they form an expression. Like y = f(x) is a general expression. This equation says that y is a function of x, so the variable x is an independent variable and the variable y is dependent on x.

y = f(a, b, c), here also y is a function of a,b and c. So the variable y is dependent whereas a, b and c are independent variables.

Now the expression properly is a mathematical term which uses variables and the constants with different mathematical operations.

Example-

x

2

x + 2

y + 8

x + y

2x + 9z etc.

Now we move towards the equations.

An equation is an expression which is equal to some other constant or variable. Equations are used to find the values of the given variables. The word problems in the maths are generally solved by forming equations.

For Example-

2 = 2

17 = 2 + 15

x = 7

7 = x

t + 3 = 8

3 × n +12 = 100

are the equations.

Now since we know what an equation is, so we now will try solving the given equation.

This is done by two methods either by solving the given equation to bring the value of the variable or by substituting the different values of variable that is the hit and trial method.

For Example

Solve 4z + 12 ?

this we will done by solving for z that is the first method.

To make it easy we use some general steps.

Step 1- simplify the equation to bring it in the simplest form.

The given equation is already in its simplest form.

Step 2- equate the give equation to 0 by bringing all the terms to one side .

4z + 12 = 0.

Step 3 – bring the constants on one side and leave the variables on the other side.

4z= - 12.

Step 4 – make the coefficient of the variable equal to 1.

in this case we will divide the equation by 4.

4z/4 = -12/ 4 , so our equation becomes

z = -3 .

so the result after solving 4z +12 = 0 is z = - 3.

This method though is very easy for small problems but becomes very difficult for complex problems.

So we use the hit and trial method for complex problems

For Example-

Solve 4z + 12 = 0 ?

we substitute different values of z and see if it becomes equal to 0.

so put the value of z = 0

= 12= 0

which is not possible.

So we put 1

4+12 = 0

16 = 0 which is also not possible.

Now for z = -1

- 4 + 12 = 0

8 = 0

which is also not possible.

Now we put z = - 3

so

4 x – 3 + 12 = 0

-12 + 12 = 0

0=0

so this the answer for the above gives equation.

Now you will be having one question in your mind that how will you know that which value it will satisfy and how many times we will have to perform this method.

So that is the reason we generally used the first method, and avoid the hit and trial method. The method is only used when the equations formed by the first method becomes very complex.

Let's solve some word problems.

1. The sum of my age and 20 equals 40

to solve this problem, we will first form an equation. Let my age be x, so according to the problem

x + 20, that is the sum of my age plus 20, and this is equal to 40

so the equation is x + 20 = 40 .

so now by using the steps for solving a given equation. We see that the value of x = 20.

2. The difference between my age and my younger sister's age, who is 11 years old, is 5 years.

So in this problem let my age by y, my sister's age is 11 and the difference between our age is 5.

y – 11 = 5 is the equation.

Now by solving the equation by the general steps.

We get the value of y = 6.

so my age is 16.

What we have learned about the equations till now is all about the linear equations that is the equations with exponent as 1.

Now we move to the quadratic equations, those equations which have 2 as their variable powers are known as the quadratic equation. We have to keep one thing in mind before solving the equation and it is that if the equation is linear then there will be one solution for the variable, if quadratic then 2 solutions, if three than 3 solutions and so on.

For Example.

is a Quadratic equation.

The quadratic equation is solved by using a general formula known as the quadratic formula.

This is the general formula that we use to solve the quadratic equations. Now the two solutions will be

Let's now solve some problems on quadratic equations.

For Example

Solve x² + 5x + 6 = 0.

we use the quadratic formula to solve the above equation.

Here a = 1, b= 5 and c = 6.

so we will put it in the formula and get two values of x.

after solving the values x comes out to -2 and -3.

Now we move to the inequalities.

Inequality is used to compare the two sides of the given equation. It is represented by using two symbols, “<” for less than and “>” for greater than. Let's understand this with the help of some notations.

x < y = this means that x is lesser than y.

x > y = this means that x is greater than y.

The other notations used are

1. ≤ for less than or equal to.
2. ≥ for greater than or equal to.
3. ≠ for not equal to.
4. <<much lesser than.
5. >> much greater than.

These notations are not used very commonly, but they do sometimes come in the questions.

There are some properties of inequalities that we need to know.

1. transitivity .

If a > b and b > c then we can say that a > c. Or we can write it as if a < b and b < c than a < c.

2. addition and subtraction property.

If a < b then, a + c < b + c or a – c < b – c.

3. for multiplication.

If a < b then, ac < bc or a/c < b/c.

4. the inverse property.

If a < b then -a > -b.

if a > b then -a < -b.

5. the multiplication inverse.

If a < b then 1/a > 1/b. And if a > b then 1/a < 1/b.

The inequalities are generally solved graphically.

For Example.

1.Solve x + 3 < 2 ?

so we solve the inequality by taking the constant on one side and variables on the other.

= x < -1

so this is the answer for the equation.

This tells us that all the region from -1 to negative infinity is the answer.

2.. solve 2 – x > 0.

for this we will use same procedure.

Which gives x < 2

So all the region from 2 to negative infinity is the answer.

3. Solve: -2y <-8 ?

so by using the reversing property, the equation becomes.

2y > 8

y > 4.

This tells that all the region above 4 is the answer.

We can plot this one also like the others.

Now moving towards much more complex examples.

4. Solve -2 < (6-2x)/3 < 4?

we can see that in this we have two inequalities at the same time.

We will solve it by taking one inequality at a time.

So now we have two inequalities that is, -6< 6 – 2 x and 6 – 2 x < 12.

Solving the fist one we get the answer for x < 6 and the second inequality gives us x > -3.

so the final answer is -3 < x < 6.

So that's all for today and I hope that you will have no problem in solving any of the questions from Variables, expressions, equations and inequalities.

How to tackle eighth standard Algebra

Friends we all know about Numbers and how to combine them using various operations like addition, subtraction, multiplication and division. This area of study is what we call as Arithmetic. The more advanced area of study of Algebra are distinct from arithmetic in which addition or basic operations to specific numbers involves entities, what we called as variables that have no particular value or we specify it as unknown value. Variables in common are generally denoted by upper or lower case letters. Some of the basic examples to show variables are “3a” , “x”, “ 7 + d” here a, x and d are variables having unknown values.

Let’s talk about algebraic expression in mathematics. An algebraic expressions is basically a collection of letters and numbers combined together to form an expression. These letters and numbers are combined together by the four basic arithmetic operations. Some of the examples of algebraic expressions are 7a, 7a + b, 7a – 4b, a / (a + b), a², (a + b)²
Here all the numbers used in algebraic expression are called constants and all the letters used in the above expressions like “a” and “b” are variables. If the expression comes with no variables then the algebraic expression is stated as arithmetic expressions. For example 4 + 7 / 6

Variables are generally used to explain the general situations or real time situations and they can also be used to solve problems that in anyway would be very difficult or even impossible to solve. Whenever we are going to solve algebraic expressions or any kind of word problems, we will see the use of these applications.

Now I am going to discuss about another topic which is very important that is an equation. In simplest mathematical manner we can say that an equation is an allegation in which two algebraic expressions used are equal. It is further stated in two different ways that are:
The first case: the given equation is true for all the values of the variables. In such kind of situation, equation is called an identity. The example to show an identity or we can say that values of variables which is true for the given equation. Most simple example is
x + y = y + x
It is also known as commutative law of addition. Another most common and well known identity is the first binomial formula or algebraic formulas like
(a + b)² = a² + 2ab + b²

The second case: In this situation the equation is true for some values of the variables. In such kind of problems or equations what we need to do is to identify those values of the variables for which the equation is true. This process is known as solving the equation. Moving forward we will study how to solve equations but for instance let’s take an example to understand the case in better manner
3a + 1 = 7
in this case obviously
a = 2 is the solution.

The given example is of a linear equation. We will further study about quadratic equations, but for know the example of a quadratic equation is a² – a – 2 = 0. If we are going to find out values for variables then, this equation has the two results that are:
a = -1 or a = 2

Now I am going in deep with the topic so the next topic is, How to evaluate an algebraic expression? To evaluate or to explore an algebraic expression refers to substitute or place specific values (desired values) for its variables. Let’s take a simple example of an algebraic expression to understand it better:
algebraic = 2a + 1

Now we are going to evaluate the given algebraic expressions. Now what we need to do is to try various values of variable which satisfy the given algebraic expression. Lets take a = 3, which provides us: algebraic = 2 x 3 + 1 = 7. we can say that the value of algebraic at a = 3 is 7. let’s take an example which is little tougher or  a bit complex then the above one
Now if we use above algebraic and put value of a = 2b + 1, where b is another variable which gives
Algebraic = 2 x (2b + 1) + 1 = 4b + 2 + 1 = 4b + 3
Now we can further solve this by substituting different values for variable b.

Before proceeding further towards inequalities let’s talk about equivalent expressions: Two expressions are equivalent if their values are equal for all possible evaluations of the two expressions. In other words presenting them with an equality sign between the expressions gives an identity.

Now I am going to talk about eighth grade linear equations topic:
In simple mathematical manner we can say that any equation that when graphed produces a straight line, then the equation is called as Linear Equation or we can say that any equation is a linear if it can be written in the linear form : ax = b. Here x is the variable, a and b are the constants. The common form of a linear equation in the two variables like x and y is
y = mx + b
where x and y are two variables and m and b are two constants.

The constant m determines the slope or gradient of that line and the constant b shows the point at which line crosses the Y-axis. Constant b is also known as Y-Intercept.
There are three possible solutions for the linear equations:
Unique Solutions: if a not equal to b then only possible solution: x = b/a
No Solution: If a = 0 and b is not equal to 0 then it has no solution.
Infinite Solutions or many solutions: if a = 0 and b = 0. In this solution 0x = 0 and there are infinite solutions for all the values of x.
Let’s take an example : 2a + 3 = a + 5
For solving this we need to subtract 3 from both the sides 2a = a + 2 that will result in a = 2.
The most important thing to understand is that, for solving a linear equation we need to recognize that the equation is linear or non linear and if linear then convert it to the simplest form. Lets take an example to elaborate it well:
(a – 2)(a – 3) = (a + 1)(a + 2)
what we need to do is to apply distributive property to both the sides of the equation
axa – ax3 – ax2 + 2x3 = axa + 2xa + 1xa + 1x2
on further simplifying this we get
a² – 5a + 6 = a² + 3a + 2
This is what we get, we can further simplify it by subtracting a² from both the sides.
Now after telling the basic concept behind linear equations, I am going to discuss about inequalities for class eighth. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. The most important thing to understand is the use of inequality symbols. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≤ less than or equal to etc.

In eighth standard we learn about linear inequality. So what linear inequality means is the first query comes in our mind. A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything on One side of a line on a graph. In mathematics a linear inequality is an inequality which involves a linear function.

The most important thing is to understand, how the inequality sign reverse when negative value comes. So to understand linear inequality, take an example: How to solve a compound inequality and graph the solutions?
Example: -6 < 2x - 4 < 12
-6 < 2x - 4 < 12
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8
To graph the following equation, you put an open circle or we can say mark it by a dot on the point (-1,0) and then you put an open circle on the point (8,0).Then draw a line between the two.

We need to be careful while solving inequalities, as they are harder to solve than equations and require more attention. You can multiply an equation by a positive or negative number and get an equivalent equation. But while solving Inequalities remember this: when multiplying it by a negative number, you need to change sign of the inequality.