Pair of Linear Equations In Two Variables
Category : 10th Class
Pair of Linear Equations in Two Variables
Note: A pair of linear equations in two variables is called a system of simultaneous linear equations.
Note: (i) A pair of values (x, y) that satisfies an equation is called its solution. (ii) Every linear equation in two variables has an infinite number of solutions.
(iii) Every solution of a linear equation is a point on the line representing it.
(i) Graphical method: The graph of a linear equation is a straight line. The graph of a pair of linear equations in two variables is represented by two lines.
(a) If the two lines intersect at a point, then the pair of linear equations has a unique solution (the point) and is said to be consistent.
(b) If the two lines coincide, then the pair of linear equations has infinitely many solutions (each point on the line being a solution), and is said to be dependent or consistent.
(c) If the two lines are parallel, then the pair of linear equations has no solution (no common point) and is said to be inconsistent.
In other words, there are three types of solutions of a pair of linear equations in two variables:
(a) Unique solution
(b) Infinitely many solutions
(c) No solution.
Note: Graphical method does not give an accurate answer as error is likely to occur in reading the coordinates of a point.
(ii) Algebraic methods: To obtain accurate result of solution of simultaneous linear equations, algebraic methods are used.
There are three algebraic methods to learn for this year.
(a) Method of elimination by substitution.
(b) Method of elimination by equating the coefficients.
(c) Method of cross multiplication.
Consider the pair of linear equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0;\,{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]
Conditions |
Types of lines |
No. of Solutions |
Type of equation |
\[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\] |
Intersecting lines |
1 (Unique solution) |
Consistent |
\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\] |
Coincident lines |
Infinitely many |
Dependent (or) consistent |
\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}\] |
Parallel lines |
No Solution |
Inconsistent |
When\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\]the system of homogeneous equations has infinitely many solutions and the system is consistent.
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