RBSE Class 10 Maths Notes Chapter 3 Pair of Linear Equations in Two Variables

These comprehensive RBSE Class 10 Maths Notes Chapter 3 Pair of Linear Equations in Two Variables will give a brief overview of all the concepts.

RBSE Class 10 Maths Chapter 3 Notes Pair of Linear Equations in Two Variables

Important Points

1. An equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables x and y We often denote the condition a and b are not both zero by a² + b² ≠ 0.

2. Every solution of the equation is a point on the line representing it.

3. Two linear equations in two variables are called a pair of linear equations in two variables. The general form of a pair of linear equations in two variables x and y is a_xx + b_xy + c_x = 0 and a₂x + b₂y + c₂ = 0
Where a_x, a₂, b_x. b₂. c_x and c₂ are all real numbers such that \(a_{1}^{2}+b_{1}^{2} \neq 0, a_{2}^{2}+b_{2}^{2}\) ≠ 0.

4. A pair of linear equation in two variables can be represented, and solved by the

• graphical method
• algebraic method

5. Graphical Method—The graph of a pair of linear equation in two variables represents two lines which show the following properties.

• If the lines intersect at a point then that point gives the unique solution of the two equations. In this case, the pair of equation is consistent.
• If the lines are coincident, then they have infinitely many solutions. Each point on the line is a solution. In this case, the pair of equation is dependent (consistent).
• If the lines are parallel, then the pair of equation has no solution. In this case, the pair of equation is inconsistent.

6. Algebraic Methods – We know the following methods for finding the solutions of a pair of linear equations.

• Substitution Method – In this method we substitute the value of one variable by expressing it in terms of the other variable to solve the pair of linear equations. This is why this method is known as the substitution method.
• Elimination Method – In this method we eliminate any one variable and get a linear equation in one variable. Solving that linear equation we find the value of any one variable. Substituting that value in any one equation the value of the second variable can be determined.
• Cross-Multiplication Method – In this method we write both the equation such that their right hand sides are zero. Let the equations be
  a₁x + b₁y + c₁ = 0
  and a₂x + b₂y + c₂ = 0

\(\frac{x}{b_{1} c_{2}-b_{2} c_{1}}=\frac{y}{c_{1} a_{2}-a_{1} c_{2}}=\frac{1}{a_{1} b_{2}-b_{1} a_{2}}\)

We can write the above in the determinant form as follows:

According to this diagram the product of the numbers with arrows upwards (T) is subtracted from the product of the numbers with arrows downwards and is written.

7. If the given linear equations a₁x+ b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 represent a pair of linear equations, then the following situations can arise :

(i) \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)
In this case lines intersect at a point and this point gives the unique solution of two equations. In this case, the pair of equation is consistent.

(ii) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)
In this case lines are parallel and the pair of equations has no solution. In this case, the pair of equations is inconsistent.

(iii) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
In this case lines coincide each other and the pair of equations has infinitely many solutions. In this case, the pair of equations is dependent and consistent.

8. There are several situations which can be mathematically represented by two such equation that are not linear to start with, but by altering them we can reduce them to a pair of linear equations.

9. Applications of Pairs of Linear Equations—

• We represent those unknown quantities which are to be determined by variables x and y.
• We convert the conditions given in the form of words in the problem into simultaneous equations in two variables x and y.
• Solving we find out the solution of the problem.

10. Solution of simultaneous equation by graph —

Graph — It is a figure which shows the lines and the curves in the form of set of points on the graph paper and determines the relation between two variables.

To draw the graph of a given relation (between the variables) we draw just in the middle of the graph paper a dark horizontal line on XOX’ axis and a dark vertical line on YOY’ axis. The horizontal line XOX’ is called the .v-axis. The vertical line YOY’ is called the y-axis.

Thus by drawing these two lines the graph paper is divided into four parts which are respectively called I, II, III, IV quadrants. In the first quadrant all these coordinates come which are both (+, +), in the II quadrant negative coordinates of x and positive coordinates of y are marked, in the III quadrant both the negative coordinates find the place and in IV quadrant the coordinates with positive Value of x and negative values of y find place.

The point where the lines XOX’ and YOY’ intersect each other is called origin and the coordinates of origin are always (0, 0).

Coordinates—To locate these on the graph paper.
All those ordered pairs which represent the distance of any point from .Y-axis and y- axis are called coordinates.

11. Representation and solution of simultaneous equations – Thus when the coordinates obtained from simultaneous equation in two variables are marked and joined them we alwaxs get a figure of two straight lines which, in general, intersect each other or intersect each other when produced.

12. Point of Intersection – Find the point where the graphs of the two straight lines intersect each other. From this point we draw perpendiculars on .Y-axis and y-axis of the graph paper. The distances, which these perpendiculars cut on XOX’ axis and YOY’ axis will be the solution of the equations.