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.