Summary
In this chapter, we have studied the following points: 1. Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations 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 +â‰ 0, +â‰ 0. Â2. A pair of linear equations in two variables can be represented, and solved, by the: (i) graphical method (ii) algebraic method
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent. (ii) If the lines coincide, then there are infinitely many solutions â€” each point on the line being a solution. In this case, the pair of equations is dependent (consistent). (iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.
(i) Substitution Method (ii) Elimination Method (iii) Crossmultiplication Method 5. If a pair of linear equations is given by 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 : (i) : In this case, the pair of linear equations is consistent. (ii) : In this case, the pair of linear equations is inconsistent. (iii) : 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 pair of linear equations.
