CBSE Class 10 Mathematics Chapter 3 - Pairs of linear equations in two variables - Exercise 3.1

 CBSE Class 10 Mathematics  Chapter 3 - Pairs of linear equations in two variables - Exercise 3.1

Q1.	Form the pair of linear equations in the following problems, and find their solutions graphically.
(i)	10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
Ans.	Assume that: Number of Girls student are x Numbers of Boys student are y As per question, first condition: total 10 students (girls + boys) participate in quiz. So, x + y = 10 ………….. (i) Second condition: The number of girls is 4 more than the number of boys, so we can write as: x = y + 4        or    x — y = 4 --------- (ii) Make a table of value of above equations. For equation (i) table of value: x 3 4 5 6 7 8 y 7 6 5 4 3 2 For equation (ii) table of value: x 3 4 5 6 7 8 y -1 0 1 2 3 4 Plot both equations on graph So, number of girls participate in quiz is 7 and boys 3.
(ii)	5 pencils and 7 pens together cost Rupees 50, whereas 7 pencils and 5 pens together cost Rupees 46. Find the cost of one pencil and that of one pen.
Ans.	Assume that: Cost of One pencil is x Cost of one pen is y As per question, first condition: 5 pencils and 7 pens together cost Rupees 50. So, 5x + 7y = 50 ………….. (i) Second condition: 7 pencils and 5 pens together cost Rupees 46, so we can write as: 7x + 5y = 46 --------- (ii) Make a table of value of above equations. For equation (i) table of value: x 0 1 2 3 4 5 y 7.1 6.4 5.7 5 4.2 3.5 For equation (ii) table of value: x 0 1 2 3 4 5 y 9.2 7.8 6.4 5 3.6 2.2 Plot both equations on graph So, the cost of one pencil (x) is rupees 3 and Pen (y) is rupees 5.

Q2.

On comparing the ratios find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0 ; 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 ; 18x + 6y + 24 = 0 (iii) 6x – 3y + 10 = 0 ; 2x – y + 9 = 0

Ans.

We know that the general form of a linear equation is ax + by + c = 0, where a, b, and c are the coefficients of x, y, and the constant term, respectively. (i) 5x – 4y + 8 = 0 ; Here a1 = 5, b1 = –4 and c1 = 8 7x + 6y – 9 = 0  ; Here a2 = 7, b2 = 6 and c2 = –9 On comparing the ratio : Because  so, given equation has a unique solution and the lines intersect at a single point. (ii) 9x + 3y + 12 = 0 ; Here a1 = 9, b1 = 3 and c1 = 12 18x + 6y + 24 = 0  ; Here a2 = 18, b2 = 6 and c2 = 24 On comparing the ratio : Because so, given equation are coincident and the lines overlap each other and are the same line and have infinite solutions. (iii) 6x – 3y + 10 = 0 ; Here a1 = 6, b1 = – 3 and c1 = 10 2x – y + 9 = 0  ; Here a2 = 2, b2 = – 1 and c2 = 9 On comparing the ratio : Because so, given equation are Inconsistent and have no solution. It’s means they are parallel.

Q3.

On comparing the ratios find out whether the following pair of linear equations are consistent, or inconsistent.

Ans.

We know that the general form of a linear equation is ax + by + c = 0, where a, b, and c are the coefficients of x, y, and the constant term, respectively. (i) 3x + 2y = 5 ; Here a1 = 3, b1 = 2 and c1 = 5 2x – 3y = 7  ; Here a2 = 2, b2 = –3 and c2 = 7 On comparing the ratio : Because  so, given equation has unique solution and pair of linear equations are consistant. (ii) 2x – 3y = 8 ; Here a1 = 2, b1 = –3 and c1 = 8 4x – 6y = 9  ; Here a2 = 4, b2 = – 6 and c2 = 9 On comparing the ratio : Because so, given equation are Inconsistent and have no solution. It’s means they are parallel. (iii)  On comparing the ratio : Because  so, given equation has unique solution and pair of linear equations are consistent. (iv) 5x – 3y = 11; Here a1 = 5, b1 = –3 and c1 = 11 – 10x + 6y = –22  ; Here a2 = –10, b2 = 6 and c2 = –22 On comparing the ratio : Because  so, given equation has infinity many solution and pair of linear equations are consistent. (v)   On comparing the ratio : Because  so, given equation has infinity many solution and pair of linear equations are consistent.

Q4.

Which of the following pairs of linear equations are consistent or inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10 (ii) x – y = 8, 3x – 3y = 16 (iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0 (iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0

Ans.

We know that the general form of a linear equation is ax + by + c = 0, where a, b, and c are the coefficients of x, y, and the constant term, respectively. (i) x + y = 5 ; Here a1 = 1, b1 = 1 and c1 = 5 2x + 2y = 10  ; Here a2 = 2, b2 = 2 and c2 = 10 On comparing the ratio : Because  so, given equation has infinity many solution and pair of linear equations is consistent. Graphically Solution: Write table of value for both equation: For equation x + y = 5 x 0 1 2 3 4 5 y = 5 — x 5 4 3 2 1 0 For equation 2x + 2y = 10 x 0 1 2 3 4 5 y = 10 — 2x/2 5 4 3 2 1 0 Plot graph: As shown in graph pair of linear equation is consistent. (ii) x – y – 8 = 0 ; Here a1 = 1, b1 = – 1 and c1 = – 8 3x – 3y – 16 = 0 ; Here a2 = 3, b2 = – 3 and c2 = – 16 On comparing the ratio : Because  so, given equation has no solution and pair of linear equations is inconsistent. (iii) 2x + y – 6 = 0 ; Here a1 = 2, b1 = 1 and c1 = – 6 4x – 2y – 4 = 0 ; Here a2 = 4, b2 = – 2 and c2 = – 4 On comparing the ratio : Because so, given equation has unique solution and pair of linear equations is consistent. Graphically Solution: Write table of value for both equation: For equation 2x + y = 6 x 0 1 2 3 4 5 y = 6 — 2x 6 4 2 0 -2 -4 For equation 4x – 2y = 4 x 0 1 2 3 4 5 y = 4 — 4x/-2 -2 0 2 4 6 8 Plot graph: As shown in graph pair of linear equation is consistent. (iv) 2x – 2y – 2 = 0 ; Here a1 = 2, b1 = – 2 and c1 = – 2 4x – 4y – 5 = 0 ; Here a2 = 4, b2 = – 4 and c2 = – 5 On comparing the ratio : Because so, given equation has no solution and pair of linear equations is inconsistent.

Q5.

Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Ans.

Assume that length of rectangular garden is x meters And Breadth is y meters We know that the perimeter of a rectangle is 2(length+breadth) As per question First condition is: ½×2(length + breadth) = 36 Length + breadth = 36         (After simplification) So, we can write it as- x + y = 36     ………………… (i) Second condition is: the length is 4 meters more than the width, so we can write x = 4 + y or x —y = 4 ……………………(ii) Write table of value for both equation: For equation (i) X 12 20 y = 36 — x 24 16 For equation (ii) x 0 4 y = x — 4 -4 0 Plot graph: As shown in graph, equation intersect at (20,16) So, length of garden is 20 meters and breadth is 16 meters.

 

Q6.

Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines (ii) parallel lines   (iii) coincident lines

Ans.

(i) For equation 2x + 3y – 8 = 0  Here a1 = 2, b1 = 3 and c1 = – 8 For intersecting lines condition (ii) apply, so  a2 = 1, b2 = 1 and c2 = 1 Another linear equation is : x + y + 1 = 0 (ii) For equation 2x + 3y – 8 = 0  Here a1 = 2, b1 = 3 and c1 = – 8 For parallel lines condition (i) apply, so  a2 = 4, b2 = 6 and c2 = 1 Another linear equation is : 4x + 6y + 1 = 0 (iii) For equation 2x + 3y – 8 = 0  Here a1 = 2, b1 = 3 and c1 = – 8 For coincident lines condition (iii) apply, so  a2 = 4, b2 = 6 and c2 = – 16 Another linear equation is : 4x + 6y – 16 = 0

 

Q7.	Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
Ans.	Write table of value for both equation: For equation x – y + 1 = 0 x 0 -1 y 1 0 For equation 3x + 2y – 12 = 0 x 0 4 y 6 0 Plot graph: As per graph required triangle formed by these line with vertices (2,3), (-1,0) and (4,0)