TS 10th Chapter 4 Pair of Linear Equations in Two Variables TS 10th Chapter 4 Pair of Linear Equations in Two Variables, Parallel, coincident, and intersecting lines. Dependent and independents lines An equation of the form ax + by + c = 0 where a, b, c are real numbers and a2 + b2 ≠ 0 is called a linear equation in two variables x and y. Two linear equations in the same variables are called a pair of linear equations in two variables a1x + b1y + c1 = 0; a2x + b2y + c2 = 0 are the pair of linear equations in two variables in x and y. Solution of Pair of linear equations in two variables: For linear equations in two variables, there are infinitely many solutions. Ex: x + y = 10 x =7, y = 3; x = 6, y = 4; x = 1, y = 9; x =2, y = 8; x=3, y = 7 like we have infinitely many solutions. ⇰ For finding exact values of x and y we have to know two linear equations. ⇰ A pair linear equations in two variables solved by four methods Graphical method Substitution method Elimination method Cross – Multiplication method Solving the pair of linear equations in two variables by using graphical Method: 1. 2x + y −5 = 0, 3x – 2y − 4 = 0 After plotting the points in the above tables in Cartesian plane, we observe that two straight lines intersect at the point (2, 1) There is only one solution for this pair of linear equations in two variables. These equations are known as consistent pair of linear equations and they have a unique solution. 2. 2x – 3y = 15; 4x – 6y = 9 After plotting the points in the above tables in Cartesian plane, we observe that two straight lines are parallel There is no solution for this pair of linear equations in two variables. These equations are known as inconsistent pair of linear equations and they have no solution. 3. 3x + 4y = 2; 6x + 8y = 4 After plotting the points in the above tables in Cartesian plane, we observe that two straight lines are coincide There are infinitely many solutions for this pair of linear equations in two variables. These equations are known as consistent pair of linear equations and they have infinitely solution. Consistent and inconsistent: If the system of equations has a solution, then they are consistent. If the system of equations has no solution, then they are inconsistent. The relationship between coefficients and the nature of the equation system: Examples: 1. Draw the graph of the following pair of linear equations in two variables and find their solution from the graph 3x – 2y = 2 and 2x + y = 6 The two lines intersect at the point (2, 2) ∴ solutions is x = 2, y = 2 2. Draw the graph of the following pair of linear equations in two variables and find their solution from the graph x – 2y = –1 and 2x – y – 4 = 0 The two lines intersect at the point (3, 2) ∴ solutions is x = 3, y = 2 3. Represent the solution of linear equation graphically x – 2y = –3 and 2x + y = 4 The two lines intersect at the point (1, 2) ∴ solutions is x = 1, y = 2 Word problems: 1. Neha went to a sale to purchase some points and skirts. When her friend asked her how many of each she had bought, she answered “The number of skirts is two less than twice the number of points purchased. Also, the number of number of skirts is four less than four times the number of pants purchased”. Help her friend to find how many pants and skirts Neha bought. Sol: Let the number of points = x and the number of skirts = y the number of skirts is two less than twice the number points purchased ⟹ y = 2x – 2 the number of number of skirts is four less than four times the number of pants purchased ⟹ y = 4x – 4 The two lines intersect at the point (1, 0) ∴ solution is x = 1, y = 2 No. of Points = 1 and no. of skirts = 0 2. 10 students of a class X took part in a Mathematics quiz. If the no. of girls is 4 more than the no. of boys, then find the no, of boys and no. of girls who took part in the quiz. Sol: let the no. of boys = x No. of girls = y Total no. of students = 10 ⟹ x + y = 10 The no. of girls is 4 more than the no. of boys ⟹ y = x + 4 ⟹ x – y = – 4 The two lines intersect at the point (3, 7) ∴ solution is x = 3, y = 7 No. of boys = 3 and no. of girls = 7 3. Half the perimeter of a rectangular garden, whose length is 4m more than its width is 36 m. Find the dimensions of the garden. Sol: let the width of garden = x m. Length of garden = y m. Length of garden is 4m mote than its width ⟹ y = x + 4 x – y = – 4 half of the perimeter of rectangular garden is 36 m. ⟹ x + y = 36 ⟹ x – y = – 4 The two lines intersect at the point (16, 20) ∴ solution is x = 16, y = 20 Width of garden = 16 m and Length of garden = 20 m 4. The area of a rectangle gets reduced by 80
