CBSE Class 10 Mathematics Pair Of Linear Equations In Two Variables

CBSE Class 10 Mathematics - Pair of Linear Equations in two Variables. Learning the important concepts is very important for every student to get better marks in examinations. The concepts should be clear which will help in faster learning. The attached concepts made as per NCERT and CBSE pattern will help the student to understand the chapter and score better marks in the examinations.

Class X

Chapter 3: Pair of Linear Equations in two Variables

Chapter Notes

Top Definitions

1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.

2. Two linear equations in same two variables x and y are called pair of linear equations in two variables.

3. The solution of pair of linear equations a₁x+b₁y+c₁= 0 and a₂x+b₂y+c₂= 0 is the ordered pair (x, y) which satisfies both the equations.

Top Concepts

1. A linear equation in two variables is represented geometrically by a straight line.

2. Each solution of a linear equation in two variables, ax + by + c = 0, corresponds to a point on the line representing the equation and vice versa.

3. The general form of a pair of linear equations in two variables is
a₁x b₁y xc₁0=0

a₂ x b₂yxc₂0=0

where a₁, a₂, b₁, b₂, c₁, c₂ are real numbers, such that a²₁ +b²₁≠0,a²₂ b²₂≠0

4. A system of linear equations in two variables represents two lines in the plane. For two given lines there could be three possible cases: 

(i) Intersecting lines, lines may intersect at a point.

(ii) Parallel lines.

(iii) Overlapping or coincidental to each other,

5. If the lines intersect at a point, then that point gives the unique solution of the system of equations. In this case system of equations is said to be consistent.

6. If the lines coincide (overlap), then the pair of equations will have infinitely many solutions. System of equations is said to be dependent and consistent.

7. If the lines are parallel, then the pair of equations has no solution. In this case pair of equations is said to be inconsistent.

8. System of equations can be solved using Algebraic and graphical.

9. Graphical method can be used to obtain the solution of a system of equations but it has its limitations in cases where the solution is non-integral.

10. Steps to be followed while using the method of substitution for solving linear equations in 2 variables:

Step 1: Find the value of one variable, say y in terms of the other variable. i.e. x from either equation, whichever is convenient.

Step 2: Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e. in terms of x, which can be solved.

Step 3: Substitute the value of x (or y) obtained in step2 in the equation used in step1 to obtain the value of the other variable.

Step 4: The values of x and y so obtained are the coordinates of the solution of system of equations.

10. There could be three possibilities on substituting the variable in the other equation:

(i) Equation reduces to a linear equation in one variable x which can be solved to get the value of x and then y.

(ii) Equation reduces to a true equation involving no variable, then the given pair of equation has infinitely many solutions

(iii) Equation reduces to false equation involving no variable then the given pair of equation has no solution.

11. Steps to be followed in Elimination Method of solving simultaneous linear equations:

Step 1: First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.

Step 2: Then add or subtract one equation from the other so that one variable gets eliminated. If you get an equation in one variable, go to step 3.

If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.

If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e. it is inconsistent.

Step 3: Solve the equation in one variable (x or y) so obtained to get its value.

Step 4: Substitute this value of x (or y) in either of the original equations to get the value of the other variable.

12. Equations which are not linear but can be reduced to linear form by some suitable substitutions are called equations reducible to linear form.

13. The speed of the boat downstream is the sum of speed of boat in still water and speed of the stream.

14. The speed of the boat upstream is the difference of speed of boat in still water and speed of the stream.

15. Reduced equation can be solved by any of the algebraic method (substitution, elimination or cross multiplication) of solving linear equation.

Top Formulae

Top Diagrams

1. Intersecting line having unique solution.

2. Parallel lines having no solution.

3. Coincident lines having infinitely many solutions.

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