Solvability of Linear Simultaneous Equations

To understand the condition for solvability of linear simultaneous equations in two variables, if linear simultaneous equations in two variables have no solution, they are called inconsistent whereas if they have solution, they are called consistent.


In the method of cross-multiplication, for the simultaneous equations, 

a₁x + b₁y + c₁ = 0 --------- (i) 

a₂x + b₂y + c₂ = 0 --------- (ii) 

we get: x/(b₁ c₂ - b₂ c₁) = y/(a₂ c₁ - a₁ c₂) = 1/(a₁ b₂ - a₂ b₁)

that is, x = (b₁ c₂ - b₂ c₁)/(a₁ b₂ - a₂ b₁) , y = (a₂ c₁ - a₁ c₂)/(a₁ b₂ - a₂ b₁) --------- (iii) 

Now, let us see when the solvability of linear simultaneous equations in two variables (i), (ii) are solvable. 

(1) If (a₁ b₂ - a₂ b₁) ≠ 0 for any values of (b₁ c₂ - b₂ c₁) and (a₂ c₁ - a₁ c₂), we get unique solutions for x and y from equation (iii) 


For examples:

7x + y + 3 = 0 ------------ (i)

2x + 5y – 11 = 0 ------------ (ii)

Here, a₁ = 7, a₂ = 2, b₁ = 1, b₂ = 5, c₁ = 3, c₂ = -11

and (a₁ b₂ - a₂ b₁) = 33 ≠ 0 from equation (iii)

we get, x = -26/33 , y = 83/33

Therefore, (a₁ b₂ - a₂ b₁) ≠ 0, then the simultaneous equations (i), (ii) are always consistent.

(2) If (a₁ b₂ - a₂ b₁) = 0 and one of (b₁ c₂ - b₂ c₁) and (a₂ c₁ - a₁ c₂) is zero (in that case, the other one is also zero), we get,

a₁/a₂ = b₁/b₂ = c₁/c₂ = k (Let) where k ≠ 0

that is, a₁ = ka₂, b₁ = kb₂ and c₁ = kc₂ and changed forms of the simultaneous equations are

ka₂x + kb₂y + kc₂ = 0

a₂x + b₂y + c₂ = 0

But they are two different forms of the same equation; expressing x in terms of y, we get

x = - b₂y + c₂/a₂

Which indicates that for each definite value of y, there is a definite value of x, in other words, there are infinite number of solutions of the simultaneous equations in this case?




For examples:

  7x +   y + 3 = 0

14x + 2y + 6 = 0

Here, a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2

Actually, we get the second equation when the first equation is multiplied by 2. In fact, there is only one equation and expressing x in term of y, we get:

x = -(y + 3)/7

Some of the solutions in particular:

simultaneous equations in two variables, simultaneous equations


(3) If (a₁ b₂ - a₂ b₁) = 0 and one of (b₁ c₂ - b₂ c₁) and (a₂ c₁ - a₁ c₂) is non-zero (then the other one is also non-zero) we get,

(let) k = a₁/a₂ = b₁/b₂ ≠ c₁/c₂

That is, a₁ = ka₂ and b₁ = kb₂

In this case, the changed forms of simultaneous equations (i) and (ii) are

ka₂x + kb₂y + c₁ = 0 ………. (v)

a₂x + b₂y + c₂ = 0 ………. (vi)

and equation (iii) do not give any value of x and y. So the equations are inconsistent.

At the time of drawing graphs, we will notice that a linear equation in two variables always represents a straight line and the two equations of the forms (v) and (vi) represent two parallel straight lines. For that reason, they do not have any common point.

For examples:

7x + y + 3 = 0

14x + 2y - 1 = 0

Here, a₁ = 7, b₁ = 1, c₁ = 3 and a₂ = 14, b₂ = 2, c₂ = -1

and a₁/a₂ = b₁/b₂ ≠ c₁/c₂

So, the given simultaneous equations are inconsistent.

From the above discussion, we can arrive at the following conclusions that the solvability of linear simultaneous equations in two variables

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 will be

(1) Consistent if a₁/a₂ ≠ b₁/b₂: in this case, we will get unique solution

(2) Inconsistent, that is, there will be no solution if

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ where c₁ ≠ 0, c₂ ≠ 0

(3) Consistent having infinite solution if

a₁/a₂ = b₁/b₂ = c₁/c₂ where c₁ ≠ 0, c₂ ≠ 0


 Simultaneous Linear Equations

Simultaneous Linear Equations

Comparison Method

Elimination Method

Substitution Method

Cross-Multiplication Method

Solvability of Linear Simultaneous Equations

Pairs of Equations

Word Problems on Simultaneous Linear Equations

Word Problems on Simultaneous Linear Equations

Practice Test on Word Problems Involving Simultaneous Linear Equations


 Simultaneous Linear Equations - Worksheets

Worksheet on Simultaneous Linear Equations

Worksheet on Problems on Simultaneous Linear Equations









8th Grade Math Practice

From Solvability of Linear Simultaneous Equations to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Conversion of Rupees and Paise | How to convert rupees into paise?

    Feb 12, 25 02:45 AM

    Conversion of Rupees and Paise
    We will discuss about the conversion of rupees and paise (i.e. from rupees into paise and from paise into rupees). How to convert rupees into paise? First we need to remove the point and then remove

    Read More

  2. Writing Money in Words and Figure Worksheet With Answers | All Grades

    Feb 12, 25 02:30 AM

    In writing money in words and figure worksheet we will get different types of questions on expressing money in words and figures, write the amount in short form (in figures) and write the amount in lo…

    Read More

  3. Writing Money in Words and Figure | Rules for Writing Money in Words

    Feb 12, 25 01:59 AM

    Rules for writing money in words and figure: 1. Abbreviation used for a rupee is Re. and for 1-rupee it is Re. 1 2. Rupees is written in short, as Rs., as 5-rupees is written as Rs. 5

    Read More

  4. Worksheet on Money | Conversion of Money from Rupees to Paisa

    Feb 11, 25 09:39 AM

    Amounts in Figures
    Practice the questions given in the worksheet on money. This sheet provides different types of questions where students need to express the amount of money in short form and long form

    Read More

  5. Worksheet on Measurement | Problems on Measurement | Homework |Answers

    Feb 10, 25 11:56 PM

    Measurement Worksheet
    In worksheet on measurement we will solve different types of questions on measurement of length, conversion of length, addition and subtraction of length, word problems on addition of length, word pro…

    Read More