NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.1 - Pair of Linear Equations in two variables

Q1 (i) Form the pair of linear equations in the following problems and find their solutions graphically. 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.

Answer:

Let the number of boys be x and the number of girls be y.

Now, according to the question,

Total number of students in the class = 10, i.e.

$\Rightarrow x+y=10.....(1)$

And, given that the number of girls is 4 more than the number of boys it means; $x=y+4$

$\Rightarrow x-y=4..........(2)$

Different points (x, y) satisfying equation (1)

Different points (x,y) satisfying equation (2)

Graph,

From the graph, both lines intersect at the point (7,3). That is x = 7 and y = 3, which means the number of boys in the class is 7 and the number of girls in the class is 3.

Q1 (ii) Form the pair of linear equations in the following problems and find their solutions graphically. 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen.

Answer:

Let the price of 1 pencil be x, and y be the price of 1 pen.

Now, according to the question

$5x+7y=50......(1)$

And

$7x+5y=46......(2)$

Now, the points (x,y) that satisfy the equation (1) are

And, the points (x,y) that satisfy the equation (2) are

The Graph,

From the graph, both lines intersect at point (3,5), that is, x = 3 and y = 5, which means the cost of 1 pencil is 3 and the cost of 1 pen is 5.

Q2 (i) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) $\\5x - 4y + 8 = 0; \\ 7x + 6y - 9 = 0$

Answer:

Given Equations,

$\\5x - 4y + 8 = 0 $and$ 7x + 6y - 9 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\frac{a_1}{a_2}=\frac{5}{7},\:\frac{b_1}{b_2}=\frac{-4}{6}\:and\:\frac{c_1}{c_2}=\frac{8}{-9}$

It is observed that;

$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$

It means that both lines intersect at exactly one point and have a unique solution.

Q2 (ii) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (ii) $\\9x + 3y + 12 = 0; \\ 18x + 6y + 24 = 0$

Answer:

Given Equations,

$\\9x + 3y + 12 = 0$ and $18x + 6y + 24 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{9}{18}=\frac{1}{2},\\\:\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}\: and \\\:\frac{c_1}{c_2}=\frac{12}{24}=\frac{1}{2}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

It means that both lines are coincident and have infinitely many solutions.

Q2 (iii) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (iii) $\\6x - 3y + 10 = 0; \\ 2x - y+ 9 = 0$

Answer:

Given Equations,

$\\6x - 3y + 10 = 0$ and$ 2x - y+ 9 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\frac{a_1}{a_2}=\frac{6}{2}=3,\:\frac{b_1}{b_2}=\frac{-3}{-1}=3\:and\:\frac{c_1}{c_2}=\frac{10}{9}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$

It means that both lines are parallel and thus have no solution.

Q3 (i) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (i) $3x + 2y = 5; 2x - 3y = 7$

Answer:

Given Equations,

$\\3x + 2y = 5$ and $2x - 3y = 7$

Or, $3x + 2y - 5 = 0$ and $ 2x - 3y - 7 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\frac{a_1}{a_2}=\frac{3}{2},\:\frac{b_1}{b_2}=\frac{2}{-3}\:and\:\frac{c_1}{c_2}=\frac{5}{7}$

It is observed that;

$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$

It means that the given equations have a unique solution and thus the pair of linear equations is consistent.

Q3 (iI) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (ii) $2x - 3y = 8; 4x - 6y = 9$

Answer:

Given Equations,

$\\2x - 3y = 8$ and $4x - 6y = 9$

Or, $\\2x - 3y - 8 = 0$ and $4x - 6y - 9 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\\\:\frac{b_1}{b_2}=\frac{-3}{-6}=\frac{1}{2}\:and\\\:\frac{c_1}{c_2}=\frac{8}{9}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$

It means the given equations have no solution, and thus the pair of linear equations is inconsistent.

Q3 (iii) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (iii) $\frac{3}{2}x + \frac{5}{3}y = 7; 9x -10y = 14$

Answer:

Given Equations,

$\\\frac{3}{2}x + \frac{5}{3}y = 7$ and $9x -10y = 14$

Or, $\\\frac{3}{2}x + \frac{5}{3}y - 7 = 0$ and $9x -10y - 14 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{3/2}{9}=\frac{3}{18}=\frac{1}{6},\\\:\frac{b_1}{b_2}=\frac{5/3}{-10}=\frac{5}{-30}=-\frac{1}{6}\:and\\\:\frac{c_1}{c_2}=\frac{7}{14}=\frac{1}{2}$

It is observed that;

$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$

It means the given equations have exactly one solution, and thus the pair of linear equations is consistent.

Q3 (iv) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (iv) $5x - 3y = 11; -10x + 6y =-22$

Answer:

Given Equations,

$5x - 3y = 11$ and $-10x + 6y =-22$

Or, $5x - 3y - 11 = 0$ and $-10x + 6y + 22 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{5}{-10}=-\frac{1}{2},\\\:\frac{b_1}{b_2}=\frac{-3}{6}=-\frac{1}{2}\:and\\\:\frac{c_1}{c_2}=\frac{11}{-22}=-\frac{1}{2}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.

Q3 (v) On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent (v) $\frac{4}{3}x + 2y = 8; 2x + 3y = 12$

Answer:

Given Equations,

$\\\frac{4}{3}x + 2y = 8$ and $2x + 3y = 12$

Or, $\\\frac{4}{3}x + 2y - 8 = 0$ and $2x + 3y - 12 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{4/3}{2}=\frac{4}{6}=\frac{2}{3},\\\:\frac{b_1}{b_2}=\frac{2}{3}\:\:and\\\:\frac{c_1}{c_2}=\frac{8}{12}=\frac{2}{3}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.

Q4 (i) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: $x + y = 5; 2x + 2 y = 10$

Answer:

Given Equations,

$\\x + y = 5$ and $2x + 2 y = 10$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{1}{2},\\\:\frac{b_1}{b_2}=\frac{1}{2}\:and\\\:\frac{c_1}{c_2}=\frac{5}{10}=\frac{1}{2}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.

The points (x,y) which satisfy both equations are

Q4 (ii) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: $x - y = 8; 3x - 3y = 16$

Answer:

Given Equations,

$\\x - y = 8$ and $3x - 3y = 16$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{1}{3},\\\:\frac{b_1}{b_2}=\frac{-1}{-3}=\frac{1}{3}\:and\\\:\frac{c_1}{c_2}=\frac{8}{16}=\frac{1}{2}$

It is observed that:

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$

It means the given equations have no solution, and thus the pair of linear equations is inconsistent.

Q4 (iii) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: $2x + y - 6 =0; 4x - 2 y - 4 = 0$

Answer:

Given Equations,

$\\2x + y - 6 =0$ and $4x - 2 y - 4 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\\\:\frac{b_1}{b_2}=\frac{1}{-2}=-\frac{1}{2}\:and\\\:\frac{c_1}{c_2}=\frac{-6}{-4}=\frac{3}{2}$

It is observed that;

$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$

It means the given equations have exactly one solution, and thus the pair of linear equations is consistent.

The points(x, y) satisfying the equation $\\2x + y - 6 =0$ are,

And The points(x,y) satisfying the equation $\\4x - 2y - 4 = 0$ are,

GRAPH:

As we can see, both lines intersect at point (2,2) and hence the solution of both equations is x = 2 and y = 2.

Q4 (iv) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: $2x - 2y - 2 =0; 4x - 4y -5 = 0$

Answer:

Given Equations,

$\\2x - 2y - 2 =0, \qquad\\ 4x - 4y -5 = 0$

Comparing these equations with $a_1x+b_1y+c_1=0\:and\:a_2x+b_2y+c_2=0$ , we get

$\\\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\\\:\frac{b_1}{b_2}=\frac{-2}{-4}=\frac{1}{2}\:and\\\:\frac{c_1}{c_2}=\frac{-2}{-5}=\frac{2}{5}$

It is observed that;

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$

It means the given equations have no solution, and thus the 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.

Answer:

Let $ l$ be the length of the rectangular garden and $b$ be the width.

Now, according to the question, the length is 4 m more than its width, so we can write it as $l=b+4$

Or, $l-b=4....(1)$

Also given Half Parameter of the rectangle = 36 it means $l+b=36....(2)$

Now, as we have two equations, add both equations, and we get,

$l+b+l-b=4+36$

$\Rightarrow 2l=40$

$\Rightarrow l=20$

We get the value of $l$, which is 20m

Now, putting this in equation (1), we get;

$\Rightarrow 20-b=4$

$\Rightarrow b=20-4$

$\Rightarrow b=16$

Hence, the Length and width of the rectangle are 20m and 16m, respectively.

Q6 (i) 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: intersecting lines

Answer:

Given the equation,

$2x + 3y -8 =0$

We know that the condition for the intersection of lines for the equations in the form $ a_1x+b_1y+c_1=0$ and $ a_2x+b_2y+c_2=0$ is,

$\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$

So any line with this condition can be $4x+3y-16=0$

Proof,

$\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{3}{3}=1$

Hence, $\frac{1}{2}\neq1$ it means $\frac{a_1}{a_2}\neq\frac{b_1}{b_2}$

Therefore, the pair of lines has a unique solution, thus forming intersecting lines.

Q6 (ii) 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 parallel lines

Answer:

Given the equation,

$2x + 3y -8 =0$

As we know that the condition for the parallel lines for the equations in the form $ a_1x+b_1y+c_1=0$ and $ a_2x+b_2y+c_2=0$ is,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$

So any line with this condition can be $4x+6y-8=0$

Proof,

$\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{-8}{-8}=1$

Hence, $\frac{1}{2}=\frac{1}{2}\neq1$ it means $\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}$

Therefore, the pair of lines has no solutions; thus lines are parallel.

Q6 (iii) 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: coincident lines

Answer:

Given the equation,

$2x + 3y -8 =0$

As we know that the condition for the coincidence of the lines for the equations in the form $ a_1x+b_1y+c_1=0$ and $ a_2x+b_2y+c_2=0$ is,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

So any line with this condition can be $4x+6y-16=0$

Proof,

$\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{-8}{-16}=\frac{1}{2}$

Hence, $\frac{1}{2}=\frac{1}{2}=\frac{1}{2}$ it means $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Therefore, the pair of lines has infinitely many solutions; thus lines are coincident.

Q7 Draw the graphs of the equations $x - y + 1=0$and $3x +2 y - 12=0$ . Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Answer:

Given two equations,

$x - y + 1=0.........(1)$

And

$3x +2 y - 12=0.........(2)$

The points (x,y) satisfying (1) are

And The points(x,y) satisfying (2) are,

GRAPH:

From the graph, we can see that both lines intersect at the point (2,3), and therefore the vertices of the Triangle are ( -1,0), (2,3) and (4,0). The area of the triangle is shaded with a green colour.