Pair Of Linear Equations In Two Variables

 

Definition: If two linear equations have the two same variables, they are called a pair of linear equations in two variables.

General forms of linear equations are:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Here, a1, a2, b1, b2, c1 and c2 are real numbers such that;

Graphical Method:

For a given pair of linear equations in two variables, the graph is represented by two lines.

1. If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.

2. If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).

3. If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

Algebraic Method:

To find the solutions of the pair of linear equations , the following methods can be used.

1. Substitution method

2. Elimination method

3. Cross-multiplication method

If a pair of linear equations is given by a1x + b1y + c1= 0 and a2x + b2y + c2 = 0 then following situations can arise.

Situation 1:

In this case, the pair of linear equations is consistent. This means there is unique solution for the given pair of linear equations. The graph of the linear equations would be two intersecting lines.

Situation 2:

In this case, the pair of linear equations is inconsistent. This means there is no solution for the given pair of linear equations. The graph of linear equations will be two parallel lines.

Situation 3:

In this case, the pair of linear equations is dependent and consistent. This means there are infinitely many solutions for the given pair of linear equations. The graph of linear equations will be coincident lines.

Question 1: Balram tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Represent this situation algebraically and graphically.

Solution: Let us assume Balram’s current age = x and his daughter’s current age = y

Seven years ago: Balram’s age = x - 7 and daughter’s age = y - 7

As per question;

x – 7 = 7(y – 7)

Or, x – 7 = 7y – 49

Or, x = 7y – 49 + 7

Or, x = 7y – 42

Or, 7y – x – 42 = 0 ……(1)

This equation gives following values for x and y

x

- 35

- 28

- 21

- 14

- 7

0

7

14

21

28

35

42

49

y

1

2

3

4

5

6

7

8

9

10

11

12

13

 

Three years from now:

Balram’s age = x + 3 and Daughter’s age = y + 3

As per question;

x + 3 = 3(y + 3)

Or, x + 3 = 3y + 9

Or, x = 3y + 9 – 3

Or, x = 3y + 6

Or, 3y – x + 6 = 0 ……..(2)

This equation gives following values for x and y:

x

9

12

15

18

21

24

27

30

33

36

39

42

45

y

1

2

3

4

5

6

7

8

9

10

11

12

13

The following graph is plotted for the given pair of linear equations:

Daughters’s age = 12 years and Balram’s age = 42 years