Linear Equations in One Variable Class 8 Notes Maths Chapter 2

Introduction

Linear Equation in One Variable

A linear equation in one variable is a special type of equation that has the following features:

• as the name suggests, It has only one variable, usually denoted by x, y, or z.
• now what does linear means ? it means that The variable has a power of 1, which means it is not raised to any other exponent like 2, 3, etc. 
• example x-7 = 3 is a linear equation but x² - 7 = 3 is not a linear equation, because here x is raised to the power of two.
  
  
  

Left Hand Side(LHS)=3x + 4
Right Hand Side(RHS)=10
For x= 2,

Let us take LHS and RHS
LHS= 3(2) + 4 = 6 + 4 = 10

RHS = 10
Now since, LHS = RHS

 x = 2 is the solution of the equation.

Constant

A constant is a value or number that never changes in an expression and it’s constantly the same.

Variable

A variable is a letter representing some unknown value. Its value is not fixed, it can take any value. On the other hand, the value of a constant is fixed.

For example, in the expression 3x+4, 3 and 4 are the constants and x is a variable.

Coefficient

Terms

A multiplication of constants and variables, is called an algebraic term. 

For example, 10x, 5y, 3z, x etc. are all algebraic terms

Algebraic Expressions

When we write one or more algebraic terms together, separated by plus or minus signs, is called an algebraic expression.

For example, 10x + 5y, 4y - 9 + 3z, 2x + 7 - 5, etc. are all algebraic expressions.

Algebraic Equation

Now, what if we want to compare two algebraic expressions using an equal sign? For example, what if we want to say that the amount of money you need to buy x chocolates is equal to the amount of money that is 20 rupees you have in your pocket? Then we can write an equation like this:

 10x = 20. 

Here, 10x is the algebraic expression on the left side of the equal sign, and 20 is the algebraic expression on the right side of the equal sign. This as a whole is called an equation.

Methods for solving linear equation:- 3 common methods                              

     Method 1                                                               Method 2                            

Method 3

Note:- Names of the following methods are : Method 1:- Balancing method, Method 2:- Transposing method, Method 3:- Cross multiplication method

Solving equation having variables on both sides

We have seen equations such as 2x-3 = 7, 4y = 2 or 4+3y = 7 that have linear expressions on the one side and numbers on the other side but this might not be the case always.
Both sides could have expressions with variables.
Let us look at some of the examples.

Example: Solve 2x – 4 = x + 2

We have 2x – 4 = x + 2

we get

2x = x + 2 + 4

2x = x + 6

2x – x = x + 6 – x (subtracting x from both sides)

x = 6 is the required answer.

Example : Solve 5x + 7/2 = 3/2x -14, then x= ?

Reducing Equations to Simpler Form

For example: Solve

Multiplying both sides by 6, we get

2 (6x + 1) + 6 = x – 3

12x + 2 + 6 = x –3

12x + 8 = x – 3

12x – x + 8 = – 3

11x + 8 = – 3

11x = –3 – 8

11x = –11

x = – 1

Example: Solve 5x – 2 (2x – 7) = 2 (3x – 1) + 9/2

LHS = 5x – 4x + 14 = x + 14
RHS = 6x – 2+  

Now,

x + 14= 6x +
14 - = 6x –x

So, x = 2.3

Example: Solve

Here, the given equation is not in linear forms.
On cross multiplying, we get
10(x+1) = 3(2x + 3)

So it gets converted into the linear form.

Now, 10x + 10 = 6x + 9

Transposing 10 and 6x to the other side, we get

10x – 6x = 9 – 10

4x = -1

Dividing both sides by 4, we get

Example:

Solve the equation:

$12(y-3)-4(y-7)+6(y+5)=012(y\; -\; 3)\; -\; 4(y\; -\; 7)\; +\; 6(y\; +\; 5)\; =\; 0$

Solution:

$12y-36-4y+28+6y+30=012y\; -\; 36\; -\; 4y\; +\; 28\; +\; 6y\; +\; 30\; =\; 0$

Combine like terms:

$(12y-4y+6y)+(-36+28+30)=0$

Isolate $\mathrm{y:}y$

$14y=-2214y\; =\; -22$

Divide both sides by 14 to solve for $yy$

$y=\frac{-}{\mathrm{22/}}=\frac{-}{11}y\; =\; \backslash frac\{-22\}\{14\}\; =\; \backslash frac\{-11\}\{7\}$

Final Answer:

$y=\frac{-}{\mathrm{11/}}y\; =\; \backslash frac\{-11\}\{7\}$