UP Board Solutions for Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

1.Two linear equations in the same two variables are called a pair of linear equations in two variables, or briefly, a linear pair. The most general form of a linear pair is

a_{1}x,+b_{1}y +c_{1}=0

a_{2}x +b_{2}y + C_{2} = 0

where a_{1}, a_{2}, b_{1}, b_{2}, C_{1}, C_{2} are real numbers. Such that

2.A pair of linear equations in two variables can be represented, and solved, by the : (i) graphical method

(ii) algebraic method

3.Graphical Method: The graph of the pair of linear equations in two variables is represented by a pair of lines.

(i) If the pair intersects at a point, then that point is the unique common solution of the two equations. In this case, the pair is consistent.

(ii) If the pair coincides, then it has infinitely many solutions – each point on the line being a solution. In this case, the pair is consistent (dependant).

(iii) If the two lines are parallel, then the pair list no solution, and is called inconsistent.

4.Algebraic Method: We have already taken the following methods for finding the solution(s) of a pair of linear equations: (i) Substitution Method (ii) Elimination Method (iii) Cross-multiplication Method

5.If a_{1}x +b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + C_{2} = 0, then the following situations can arise:

In this case, the pair of linear equations is consistent.

In this case, the pair of linear equations is inconsistent

In this case, the pair of linear equations is dependent and consistent.

6.There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so that they are reduced to a linear pair.

In solving such problems, we observe the following steps:

(i) Read the problem carefully and give the unknown quantities a variable name as x, y, z, etc.

(ii) Identify the variable to be determined.

(iii) Formulate the equations in terms of the variablesto be determined. (iv) Solve the equations obtained in step (iii) by using any one of the methods learnt earlier.