Hey there! Have you understood about Rational Numbers in the previous Chapter? In this chapter, you will be introduced to the linear equations in one variable. This is one of the basic chapters of Algebra and important as well. So for understanding, this chapter let’s first discuss about what these words mean.
Linear - Linear means the highest power of variables is 1. For example - x+7, x+y+5 or 2x+3y+10 are linear expressions, while x^{2}+9, x^{3}+y+8 or 3x+2y^{(½)} are non-linear expressions because the power of variables that is x or y is not equal to 1.
Equations - It is a statement of equality of two expressions. It always contain “equal to(=)” operator between two expressions. For example 3x + 7 = 0, 3x + 7 = 20 etc.
Variables - The terms in the equation whose value is not known are called as variables. Example- x and y in the equation “2x+3y+10” are variables.
So basically Linear equations in one variable deals with solving simple equations in which there is only variable (let’s say x). Let’s understand about other features of a linear equation in one variable.
In the above figure,
We have an equality sign in the center, which tells that this is an equation.
On the left of the equality sign, the expression is called the Left-Hand Side (LHS)
On the right of the equality sign, the expression is called the Right-Hand Side(RHS).
As a Class 8 maths student, you should be aware of the application of the chapter which you are reading. Let us understand the application by an example.
Suppose Ram’s mother’s age is 4 times the present age of Ram, after 5 years their ages add to 65 years. Then find their present ages? As you can see this question seems very lengthy and confusing. But, you could easily solve these types of problems by using a variable x. We will discuss the solution to the above problem later.
What you need to understand is that linear equations help us write a large problem in a simpler way. Now Let’s discuss the different types of linear equations with the help of examples.
Type 1 - Solving equations which have Linear expressions on one side and Numbers on the other side
Let’s recall the previous problem. Suppose Ram’s mother’s age is 4 times the present age of Ram, after 5 years their ages add to 65 years. Then find their present ages.
For solving this example, let’s assume the present age of Ram is x years
Age of Ram’s Mother, = 4x years
Now after 5 years, Age of Ram = x+5 years and Age of Ram’s mother = 4x+5 years
Sum of their ages after 5 years, (x+5)+(4x+5) = 65
=> 5x+10 = 65
Now subtracting 10 from both sides,
=>5x+10-10 = 65-10
=>5x = 55
Dividing both sides by 5,
=> x = 11
Age of Ram = x = 11 years
Age of Ram’s Mother = 4x = 4*11 = 44 years
Example 2- The perimeter of a rectangle is 13cm and its width is cm. Find its length.
Solution: Assume the length of the rectangle to be x cm.
The perimeter of the rectangle = 2 × (length + width)
The perimeter is given to be 13 cm. Therefore,
Hence, the Length of the rectangle is 15/4 cm
Type 2 - Solving equations having variables on both Sides
Example 1 -Let’s start with a simple example. Solve 2x-3 = x+2
Solution - We have, 2x-3=x+2
First, add 3 to both sides => 2x-3+3 = x+2+3
This will give us, 2x = x+5
Now subtracting x from both sides, 2x-x = x+5-x
This will give us, x = 5.
Example 2- The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. What can be the original number?
Solution: A two-digit can be written in the form of 10*(tens digit) + units digit. For example, 75 can be written as 10*7+5 after solving this you will get 75.
Coming back to our problem. Let’s assume the two-digit number is ab. Here a is ten’s digit and b is units digit.
Now according to question, the digits differ by 3 so we have, a = b+3………….(i)
The number will be, 10*(b+3) + b => 10b+30+b => 11b+30…….(ii)
After interchanging the digits, the tens digit will be b and units digit will be a
The interchanged number will be, 10*b+a => 10b+b+3 => 11b+3……….(iii)
Now it is said in the question that the sum of these two numbers will be equal to 143.
Adding equation (ii) and (iii), (11b+30)+(11b+3) = 143
=> 22b+33 = 143
Now subtract 33 from both sides, => 22b +33 -33 = 143 -33 => 22b = 110
Dividing it by 22, => b = 110/22 = 5.
Now a = b+3 = 5+3 = 8
Hence the two digit number is 85.
Checking the solution: On the interchange of digits, the number we get is 58. The sum of 85 and 58 is 143 as given.
Type 3 - Reducing Equations to Simpler Form
Example : Solve
Solution: For solving this equation, we will multiply both sides by 6 because 6 is the LCM of the denominators.
=> 2(6x+1) +6 = x-3
=> 12x+2+6 = x-3
=> 12x+8 = x-3
=> 12x +8 -8 = x-3-8 (subtract 8 from both sides)
=> 12x = x-11
=> 12x-x = x-11 -x (subtract x from both sides)
=> 11x = -11
=> x = -1 (divide by 11)
The use of linear equations in one variable is in their diverse applications; different problems on ages, numbers, combinations of currency notes, perimeters, and so on can be solved by using linear equations. In this chapter, there are 6 exercises with 65 questions. Below mentioned are the topics of NCERT for Class 8 Maths Chapter 2 Linear Equations in One Variable-
2.1 Introduction
2.2 Solving Equations which have Linear Expressions on one Side and Numbers on the other Side
2.3 Some Applications
2.4 Solving Equations having the Variable on Both Sides
2.5 Some More Applications
2.6 Reducing Equations to Simpler Form
2.7 Equations Reducible to the Linear Form
While preparing chapter 2 Linear equations in one variable, go through the NCERT concepts first and then attempt NCERT exercises. Having NCERT Solutions for Class 8 Science are going to be beneficial from the exam point of view.