## Introduction

In earlier classes, we have studied the linear equation in one variable and two variables. The general form of linear equation in one and two variables are ax+b=0 and ax+by+c=0 respectively where a,b and c are real number and a²+b²≠0. In class 10 we shall study about pair of linear equations in two variables definition, its solution by algebraic method, graphical method, and application-based word problem solution .so let’s know what is a simultaneous linear equation in two variables.

## Linear equation in two variable definition

Definition :

Any equation of form ax+by+c=0 is called a linear equation in two variables where a,b and c are real number and x,y are linear variable. Here is some example of a linear equation in two variable

Example:

(i) 2x+3y-5=10

(ii) x+9y=0

(iii) 3x-5y+10=0

I hope, now it is clear. So, let’s discuss some other problem

**What is the simultaneous linear equation in two variables?**

The pair of linear equations in two variables is called a simultaneous linear equation in two variables.

**What is the solution of a pair of linear equations in two variables?**

The solution of a pair of linear equations in two variables is the value of x and y which **satisfy both equations. I REPEAT IT **should satisfy both equations.

### Types of simultaneous linear equation in two variables

These equations are of two types

(i) consistent equation

(ii) In-consistent equation

**Consistent equation: **A simultaneous linear equation is said to be consistent if it has a least one solution.

**Inconsistent equation: **Equations are said to be inconsistent if it has no solution.

Note: We can’t solve all simultaneous linear equations, later we will discuss this and their condition in detail.

#### Method to solve simultaneous linear equation

There are four types of method which is used to solve any linear equation in two variable. But these methods are not applicable to inconsistent equations.

Now I am sure, you are thinking about how we can identify these types of equations ? wait we will discuss its condition later. First focus on the solution of equations.

(i)Substitution method

(ii)Elimination method

(iii)cross-multiplication method

(iv)graphical method

- Detail explanation of real number for class 10
- Polynomial for class 10
- Quadratic equation for class 10

**Substitution method**

As we understand about this method simply through its name substitution means (to put). So , we shall find the value of a variable from any of two given equations and substitute (put ) it in another equation. Let’s understand this method with the help of an example

Example: **Find the solution of equations with the help of the substitutions method.**

3x-5y=-1 …………..(i)

x-y=-1 …………….(ii)

solutions: we have two equations

3x-5y=-1 …………..(i)

x-y=-1 …………….(ii)

First, find the value of a variable from any of the two-equation. suppose I find the value of x from second equations

x-y=-1 …………….(ii)

x=-1+y ……………..(iii)

now substitute this value of x in the equation in first

3x-5y=-1 …………..(i)

3(-1+y)-5y=-1

-3+3y-5y=-1

-3-2y=-1

-2y=-1+3

-2y=2

y=-1

substitute this y=-1 in equation (iii)

x=-1+y ……………..(iii)

x=-1-1

x=-2

so x=-2 and y=-1 is the solution of these equations

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**Elimination method **

In the elimination method, we simply eliminate (equal) the value of the coefficient of the equation.

Example 8x+5y=9 ………….(I)

3x+2y=4 ……………(ii)

solution: method (i) Take the LCM of coefficients of x or y .suppose we take the coefficient of x i.e 8 and 3

LCM of 8 and 3 =24

now we have to make both coefficients of x equal to 24 by multiplication of 3 and 8 in equation (i) and (ii) respectively.

3( 8x+5y)=3(9)

24x+15y=27 ………………(iii)

8(3x+2y)=8(4)

24x+16y=32 ……………..(iv)

subtract equation (iii) and (iv)

(24x+15y)-(24x+16y)=27-32

24x+15y-24x-16y=-5

-y=-5

y=5

substitute y=5 in equation (i)

8x+5y=9

8x+5(5)=9

8x+25=9

8x=9-25

8x=-16

x=-2

so x=-2 and y=5 are solutions of the equation

**cross-multiplication method**

We can understand about it by its name that we have to multiply but in a cross.

suppose we have two equations

ax+by+c=0

a_{1} x+b_{1} y+c_{1} =0the solution of this equation will be

\frac{x}{bc_{1} -b_{2} c} =\frac{-y}{ac_{1} -a_{1} c} =\frac{1}{ab_{1} -a_{1} b}Here is one confusing thing which confuses most student i.e

you will see -1 instead of 1 in many books solution

but some book solve with 1

so, what is the difference

The difference is very simple **when the constant term of equations are LHS, we use 1**

**but when the constant term of the equations are on RHS, we use -1**

It totally depends on you, which way you prefer

**Graphical method**

We have already studied that graph of linear equation is a straight line whether it is of one variable, two variables…so on. The straight line represents every solution of equations. Let’s learn about it deeply

**Linear equation one variable**

Every linear equation in one variable straight line are parallel to coordinate axes

Example: 3x+5=0

x=-5/3

The straight line will pass through x=-5/3

whereas

Linear equation in two variable straight lines makes an angle with coordinate axes (about this we study in higher classes) .

Example: x-2y=1

x=1+2y

find every value x of corresponding to y.

y | 3 | 2 | 5 |
---|---|---|---|

x | 7 | 5 | 11 |

Here (7,3),(5,2),(11,5) and many more can be the solution of the equation.

when we will plot these points on the graph it will be a straight line.

**What happens when there will be a pair linear equation in two variables?**

obviously, there will be two straight lines. Now the position of these lines will represent the solution of the equation. Here is some condition which can be found.

Case-I When two lines are parallel i.e they never intersect each other.

In this case, there will be no common solution because lines never intersect each other

Case-II When two lines will intersect each other .there will be only one point which is common in both lines and that is also only a solution of both linear equations

Case-III When two lines are coincident lines i.e one line cover the other completely

**can you guess how many solutions have?**

Yes ,you are right .these lines will have infinitely many solution because every solution of one line is also the solution of another line.

**Let discuss about algebraic method to identify consistent and inconsistent solution**

Till now, we have studied about different types of method to solve linear pair of equation in two variable.Let learn about the condition which will help us to identify that a pair of equations are solvable or not

Earlier in graphical method ,we have learned about this but by graphical apporoach which takes lot of time to identify.So ,Let’s learn about algebraic method

Suppose ,we have a pair of equation

ax+by+c=0

a_{1} x+b_{1} y+c_{1} =0**Case-I When equations have one solution**

**Graphical –** When equations will have one solution,then lines of these equations intersect each other

**Algebraic method- \frac{a}{a_{1}} \neq \frac{b}{b_{1}} **

Example- 3x+5y+7=0

6x+15y+14=0

here a=3 {a_{1}}=6,b=5 and {b_{1}}=14

<strong>\frac{a}{a_{1}}=\frac{1}{2} , \frac{b}{b_{1}} =\frac{1}{3}

here **\frac{a}{a_{1}} \neq \frac{b}{b_{1}} **

That means it has unique.

**Case-II When equations have infinitely many solutions**

Graphical- The line of equations coincide each other i.e They completely cover each other

algebraic- \frac{a}{a_{1}} =\frac{b}{b_{1}} =\frac{c}{c_{1}}

Example- 3x+5y+7=0

6x+10y+14=0

Here a=3 b=5 c=7

a_{1} =6 ,b_{1}=10 ,c_{1}=14you can see how solution are varying

\frac{a}{a_1}=\frac{1}{2},\frac{b}{b_1}=\frac{1}{2} and \frac{c}{c_1}=\frac{1}{2}</p> <p>\frac{a}{a_1}=\frac{b}{b_1}= \frac{c}{c_1}so,equationa will have infinitely many solutions

**Case-III When equations have no solution(we can not solve pair of equations )**

** Graphical – **The lines of equations are parallel to each other

Algebraic- \frac{a}{a_{1}} =\frac{b}{b_{1}} =\frac{c}{c_{1}}