Sunday, December 16, 2018

Pair of Linear Equations in two Variables Exercise 3.2

Pair of Linear Equations in two Variables Exercise 3.2

 

Page No: 49

1. 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.

Answer

Let number of boys = x
Number of girls = y
Given that total number of student  is 10 so that
= 10
Subtract y both side we get
= 10 – y
Putting = 0 , 5, 10 we get
= 10 – 0  = 10
= 10 – 5 = 5
= 10 – 10 = 0

x      105y      05

Given that If the number of girls is 4 more than the number of boys
So that
=  + 4
Putting x = -4, 0, 4, and we get
=  - 4 + 4 = 0
=  0 + 4 = 4
=  4 + 4 = 8

x      -404y      048

Graphical representation



Therefore, number of boys = 3 and number of girls = 7.

(ii) 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 cost of pencil = Rs x
Cost of pens = Rs y
5 pencils and 7 pens together cost Rs 50,
So we get
5x + 7y = 50
Subtracting 7y both sides we get
5= 50 – 7y
Dividing by 5 we get
x = 10 - 7 y /5
Putting value of y = 5 , 10  and 15 we get
= 10 – 7 × 5/5 = 10 – 7 = 3
= 10 – 7 × 10/5 = 10 – 14 = - 4

= 10 – 7 × 15/5 = 10 – 21  = - 11

x      3-4-11y      51015

Given that 7 pencils and 5 pens together cost Rs 46
7x + 5= 46
Subtracting 7x both side we get
5= 46 – 7x
Dividing by 5 we get
= 46/5   - 7x/5
y = 9.2 – 1.4x
Putting x  = 0 , 2 and 4 we get
=  9.2 – 1.4 × 0 =  9.2 – 0 = 9.2
=  9.2 – 1.4 (2) = 9.2 – 2.8 = 6.4
=  9.2 – 1.4 (4) = 9.2 – 5.6 = 3.6

x      024y      9.26.43.6

Graphical representation



Therefore, cost of one pencil = Rs 3 and cost of one pen = Rs 5.

2. On comparing the ratios a1/a2 , b1/b2  and c1/c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

Answer

(i) 5x – 4y + 8 = 0
7x + 6y – 9 = 0

Comparing these equation with
a1x + b1y + c1 = 0
a2x + b2y + c2= 0

We get
a1 = 5, b= -4,  and c1 = 8
a2 =7, b= 6 and c2 = -9
a1/a2 = 5/7,
b1/b2 = -4/6 and
c1/c2 = 8/-9
Hence, a1/a2 ≠ b1/b2

Therefore, both are intersecting lines at one point.

(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
Comparing these equations with

a1x + b1y + c1 = 0

a2x + b2y + c2= 0
We get
a1 = 9, b1 = 3, and c1 = 12
a2 = 18, b2 = 6 and c2 = 24
a1/a2 = 9/18 = 1/2
b1/b2 = 3/6 = 1/2 and
c1/c2 = 12/24 = 1/2
Hence, a1/a2 = b1/b= c1/c2

Therefore, both lines are coincident

(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0
Comparing these equations with

a1x + b1y + c1 = 0

a2x + b2y + c2= 0

We get
a1 = 6, b1 = -3, and c1 = 10
a2 = 2, b2 = -1 and c2 = 9
a1/a2 = 6/2 = 3/1
b1/b2 = -3/-1 = 3/1 and
c1/c2 = 12/24 = 1/2
Hence, a1/a2 = b1/b c1/c2

Therefore, both lines are parallel

3. On comparing the ratios a1/a2 , b1/b2  and c1/c2 find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 3x + 2y = 5 ; 2x – 3y = 7
(ii) 2x – 3y = 8 ; 4x – 6y = 9
(iii) 3/2x + 5/3y = 7 ; 9– 10y = 14
(iv) 5x – 3y = 11 ; – 10x + 6y = –22
(v) 4/3x + 2y =8  ; 2x + 3y = 12 

Answer

(i) 3x + 2y = 5 ; 2x – 3y = 7
a1/a2 = 3/2
b1/b2 = -2/3 and
c1/c2 = 5/7
Hence, a1/a2 ≠ b1/b2
These linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(ii) 2x – 3y = 8 ; 4x – 6y = 9
a1/a2 = 2/4 = 1/2
b1/b2 = -3/-6 = 1/2 and
c1/c2 = 8/9
Hence, a1/a2 = b1/b c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii) 3/2x + 5/3y = 7 ; 9– 10y = 14
a1/a2 = 3/2/9 = 1/6
b1/b2 = 5/3/-10 = -1/6 and
c1/c2 = 7/14 = 1/2
Hence, a1/a2 ≠ b1/b2

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(iv) 5x – 3y = 11 ; – 10x + 6y = –22
a1/a2 = 5/-10 = -1/2
b1/b2 = -3/6 = -1/2 and
c1/c2 = 11/-22 = -1/2
Hence, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

(v) 4/3x + 2y =8 ; 2x + 3y = 12
a1/a2 = 4/3/2 = 2/3
b1/b2 = /3 and
c1/c2 = 8/12 = 2/3
Hence, a1/a2 = b1/b2 = c1/c2

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

4. Which of the following pairs of linear equations are consistent/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

Answer

(i) x + y = 5; 2x + 2y = 10
a1/a2 = 1/2
b1/b2 = 1/2 and
c1/c2 = 5/10 = 1/2
Hence, a1/a2 = b1/b2 = c1/c2
Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

x + y = 5

x = 5 - y

x      432y      123

And, 2x + 2y = 10
x = 10-2y/2

x      432y      123

Graphical representation



From the figure, it can be observed that these lines are overlapping each other. Therefore, infinite solutions are possible for the given pair of equations.

(ii) x – y = 8, 3x – 3y = 16
a1/a2 = 1/3
b1/b2 = -1/-3 = 1/3 and
c1/c2 = 8/16 = 1/2
Hence, a1/a2 = b1/b c1/c2

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
a1/a2 = 2/4 = 1/2
b1/b2 = -1/2 and
c1/c2 = -6/-4 = 3/2
Hence, a1/a2 ≠ b1/b2

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

2x + y - 6 = 0
y = 6 - 2x

x      012y      642

And, 4x - 2y -4 = 0
y = 4x - 4/2

x      123y      024

Graphical representation



From the figure, it can be observed that these lines are intersecting each other at the only one point i.e., (2,2) which is the solution for the given pair of equations.

(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
a1/a2 = 2/4 = 1/2
b1/b2 = -2/-4 = 1/2 and
c1/c2 = 2/5
Hence, a1/a2 = b1/b c1/c2

Therefore, these linear equations are parallel to each other and thus, have no possible solution. Hence, the pair of linear equations is inconsistent.

Page No: 50

5. 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 length of rectangle = x m
Width of the rectangle = m
According to the question,
y - x = 4 ... (i)
y + x = 36 ... (ii)
y - x = 4
y = x + 4

x      0812y      41216

y + x = 36

x      03616y      36020

Graphical representation



From the figure, it can be observed that these lines are intersecting each other at only point i.e., (16, 20). Therefore, the length and width of the given garden is 20 m and 16 m respectively.

6. Given the linear equation 2x + 3y - 8 = 0, write another linear equations in two variables such that the geometrical representation of the pair so formed is:
(i) intersecting lines

(ii) parallel lines
(iii) coincident lines

Answer

(i) Intersecting lines:
For this condition,
a1/a2 ≠ b1/b2
The second line such that it is intersecting the given line is
2x + 4y - 6 = 0 as
a1/a2 = 2/2 = 1
b1/b2 = 3/4 and
a1/a2 ≠ b1/b2

(ii) Parallel lines

For this condition,

a1/a2 = b1/b c1/c2
Hence, the second line can be
4x + 6y - 8 = 0 as
a1/a2 = 2/4 = 1/2
b1/b2 = 3/6 = 1/2 and
c1/c2 = -8/-8 = 1
and a1/a2 = b1/b c1/c2

(iii) Coincident lines
For coincident lines,
a1/a2 = b1/b2 = c1/c2
Hence, the second line can be
6x + 9y - 24 = 0 as
a1/a2 = 2/6 = 1/3
b1/b2 = 3/9 = 1/3 and
c1/c2 = -8/-24 = 1/3
and a1/a2 = b1/b2 = c1/c2

7. 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.

Answer

x - y + 1 = 0
x = y - 1

x      012y      123

3x + 2y - 12 = 0

x = 12 - 2y/3

x      420y      036

Graphical representation



From the figure, it can be observed that these lines are intersecting each other at point (2, 3) and x-axis at ( - 1, 0) and (4, 0). Therefore, the vertices of the triangle are (2, 3), ( - 1, 0), and (4, 0).

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