Pairs of Linear Equation in Two Variables · the equation we get the value of the ˚rst variable....
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Linear Equation in Two Variables Solving a Pair Of Linear Equations Consistency Of A Pair of Linear Equations Inconsistent Pair of Linear Equations Pair Of Linear Equations in Two Variables Graphical Representation and Algebraic Interpretation Of Pair Of Linear Equations An equation of the type ax + by + c = 0, where a, b, c are real numbers and a ≠ 0 and b ≠ 0. • Plot both the linear equations on the graph paper. This will give us two straight lines. • If the lines intersect at one point then it has unique solution. • If the lines are parallel then there is no solution for the pair of equations. • If the lines are coincident then there are infinitely many solutions. Substitution Method Elimination Method Cross Multiplication Method (a) In this, using first equation we find the value of one variable in terms of other and substitute in the 2 nd equation. (b) On simplification we get the value of one variable and resubstituting it in any one gives the value of other variable. (a) In this method, we multiply both the equations with some non-zero constant so as one of the variable gets eliminated. (b) This gives the value of other variable. Using this value and any one of the equation we get the value of the first variable. Two linear equations with two same variables are called pair of linear equations. Values of x and y for which equations hold true are called as its solution. The general form of these type of equations is Intersecting lines Coincident lines Parallel lines Exactly one solution (unique) No solution Infinitely many solutions For the lines represented by the equation S. No. Compare the ratios 1. 2. 3. Graphical representation Algebraic Interpretation Here, a1 b2 - a2 b1 ≠ 0 , for unique solution. Pair of linear equations with no solutions. Consistent Pair of Linear Equations Pair of linear equations with either unique solution or infinitely many solutions. a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 For equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 A linear equation in two variable has infinitely many solution. Graphical Method Algebraic Method When graphically represented, it represents a straight line and every solution is a point in this straight line. Where a1, a2, b1, b2 , c1, c2 are real numbers and + b ≠ 0 a 2 1 2 1 + b ≠ 0 a 2 2 2 2 ≠ a1 a2 b1 b2 = a1 a2 b1 b2 = c1 c2 = a1 a2 b1 b2 ≠ c1 c2 b1 c2 - b2 c1 a1 b2 - a2 b1 x = c1 a2 - c2 a1 a1 b2 - a2 b1 y = 1 2 1 2 1 2 3 4
Transcript of Pairs of Linear Equation in Two Variables · the equation we get the value of the ˚rst variable....
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Solving a Pair Of Linear Equations
Consistency Of A Pair of Linear EquationsInconsistent Pair of Linear Equations
Pair Of Linear Equations in Two Variables
Graphical Representation and Algebraic Interpretation Of Pair Of Linear Equations
An equation of the type ax + by + c = 0, where a, b, c are real numbers anda ≠ 0 and b ≠ 0.
• Plot both the linear equations on the graph paper. This will give us two straight lines.• If the lines intersect at one point then it has unique solution.• If the lines are parallel then there is no solution for the pair of equations.• If the lines are coincident then there are in�nitely many solutions.
Substitution Method
Elimination Method
Cross Multiplication Method
(a) In this, using �rst equation we �nd the value of one variable in terms of other and substitute in the 2nd equation.(b) On simpli�cation we get the value of one variable and resubstituting it in any one gives the value of other variable.
(a) In this method, we multiply both the equations with some non-zero constant so as one of the variable gets eliminated.(b) This gives the value of other variable. Using this value and any one of the equation we get the value of the �rst variable.
Two linear equations with two same variables are called pair of linear equations.
Values of x and y for which equations hold true are called as its solution.
The general form of these type of equations is
Intersecting lines
Coincident lines
Parallel lines
Exactly onesolution(unique)
No solution
In�nitely manysolutions
For the lines represented by the equation
S.No.
Compare the ratios
1.
2.
3.
Graphical representation
Algebraic Interpretation
Here, a1 b2 - a2 b1 ≠ 0 , for unique solution.
Pair of linear equations with no solutions.
Consistent Pair of Linear EquationsPair of linear equations with either unique solution or in�nitely many solutions.
a1x + b1y + c1 = 0a2x + b2y + c2 = 0
For equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
a1x + b1y + c1 = 0a2x + b2y + c2 = 0
A linear equation in two variable has in�nitely many solution.
Graphical Method
Algebraic Method
When graphically represented, it represents a straight line and every solution is a point in this straight line.
Where a1, a2, b1, b2 , c1, c2 are real numbers and + b ≠ 0 a 21
21 + b ≠ 0 a 2
222
≠a1a2
b1b2
=a1a2
b1b2 = c1
c2
=a1a2
b1b2 ≠ c1
c2
b1 c2 - b2 c1a1 b2 - a2 b1x =
c1 a2 - c2 a1a1 b2 - a2 b1y =
1
2
1
2
1
2
3
4
