KC Sinha Mathematics Solution Class 9 Chapter 5 Linear Equations in Two Variable exercise 5.1

Page 5.21

Exercise 5

Type 1

Problems based on writing a linear equation in two variables x and y in the form ax-by+c=0 and problems on writing a given statement as a linear equations in two variables

WORKING RULE :

Use the following information wherever is required :

1. In order to write a given statement as a linear equation in two variables , identify the two variables and let these be x and y

2. Then, write the given condition in terms of x and y . The equation thus obtained will be the required equation.

Question 1

Write each of the following equations in the Form ax + by + c = 0 and indicate the values of a , b and c in each case :

(i) 2x+3y=4.37

Sol :

⇒2x+3y-4.37=0

a=2 , b=3 , c=-4.37

(ii)

Sol :

a=1 , b= -√3 , c= -4

(iii) 4=5x-3y

Sol :

⇒5x-3y-4

a=5 , b=-3 , c= -4

(iv) 2x=y

Sol :

⇒ 2x-y-0=0

a=2 , b=-1 , c=0

Question 2

Write each of the following as an equation in two variables :

(i) x=-5

Sol :

⇒x+5=0

⇒x+0+5=0

⇒x+0y+5=0

(ii) y=2

Sol :

⇒y-2=0

0+y-2=0

0x+y-2=0

(iii) 2x=3

Sol :

⇒2x-3=0

⇒2x+0-3=0

⇒2x+0y-3=0

(iv) 5y=2

Sol :

⇒5y-2=0

0+5y-2=0

0x+5y-2=0

Type 2

Problems based on examining whether given values are solution of the given equation or not.

WORKING RULE:

Put the given values in the given equation and simplify. If the given equation is satisfied by these values , then given values will be solution of given equation otherwise they will not

Question 3

Examine whether x = 2, ,y=1 , are solutions of the following equations or not ?

(i) 2x+5y=7

Sol :

⇒On putting x=2 and y=1

⇒2×2+5×1=7

⇒4+5=7

⇒9=7

NO

(ii) 2x-3y+7=8

Sol :

⇒On putting x=2 and y=1

⇒2×2-3×1+7=8

⇒4-3+7=8

⇒11-3=8

⇒8=8

YES

(iii) 3x+4y=9

Sol :

⇒On putting x=2 and y=1

⇒3×2+4×1=9

⇒6+4=9

⇒10=9

NO

(iv) 5x-7y=3

Sol :

⇒On putting x=2 and y=1

⇒5×2-7×1=3

⇒10-7=3

⇒3=3

YES

Question 4

Are x=-1 , y=3 solutions of the following equations or not ?

(i) 2x+5y=13

Sol :

⇒On putting x=-1 and y=3

⇒2×(-1)+5×3=13

⇒-2+15=13

⇒13=13

YES

(ii) 2x-3y=-11

Sol :

⇒On putting x=-1 and y=3

⇒2×(-1)-3×3=-11

⇒-2-9=-11

⇒-11=-11

YES

(iii) 5x+3y=4

Sol :

⇒On putting x=-1 and y=3

⇒5×(-1)+3×3=4

⇒-5+9=4

⇒4=4

YES

(iv) 2x+3y=41

Sol :

⇒On putting x=-1 and y=3

⇒2×(-1)+3×3=41

⇒-2+9=41

⇒7=41

NO

Question 5

In the following equation, f‌ind the values of a , so that x=1 , y=1 is its solution:

(i) 5x+3y=a

Sol :

⇒On putting x=1 and y=1

⇒5×1+3×1=a

⇒5+3=a

⇒a=8

(ii) ax-2y=10

Sol :

⇒On putting x=1 and y=1

⇒a×1-2×1=10

⇒a-2=10

⇒a=10+2

⇒a=12

(iii) ax+4y=18

Sol :

⇒On putting x=1 and y=1

⇒a×1+4×1=18

⇒a+4=18

⇒a=18-4

⇒a=14

(iv) 7ax+3ay=20

Sol :

⇒On putting x=1 and y=1

⇒7a×1+3a×1=20

⇒7a+3a=20

⇒10a=20

⇒a=2

Type 3

Problems based on solution of linear equation in two variables.

WORKING RULE:

Use the following information whichever is required :

1. A linear equation in one variable has unique solution (one and only one solution)

2. A linear equation in two variables has inf‌initely many solutions.

3. Any number of solutions of a linear equation in two variables can be written.

For this :

(i) If linear equation in two variables be ax+by+c=0 , where a,b,c are real numbers and a≠0 , b≠0 then write y in terms of x

Here , by=-(ax+c) or

(ii) Putting arbitrary real values of x, f‌ind the corresponding values of y. For convenience, putting y = 0 , find the value of x and putting x = 0, find the value of y

These solution will be of the form x= a, y = 0 and y = b.

(iii) We can f‌ind as many solutions as we need by putting arbitrary values of x and finding corresponding values of y.

All these pair of values of x and y will be solutions of given linear equation

Question 6

Find two solutions of each of following equations :

(i) 4x+3y=12

Sol :

⇒3y=12-4x

When x=3 , =0 ;

When x=1 ,

⇒(3,0);

(ii) 2x+5y=0

Sol :

⇒5y=-2x

When x=0 , ;

When x=1 , ;

⇒(0,0)

(iii) 3y+4=0

Sol :

0x+3y+4=0

⇒3y=-4-0x

When x=0 , ;

When x=1 , ;

Question 7

Find at least three solutions of the following equations :

(i) 5x+3y=4

Sol :

⇒3y=4-5x

When x=-1 ,

When x=2 ,

When x=-4 ,

⇒ x= -1, y= 3; x= 2, y= -2; x= -4, y= 8

(ii) 2x-3y=-11

Sol :

⇒-3y=-11-2x

When x=-4 ,

When x=-1 ,

When x=2 ,

⇒ x= -4, y= 1; x= -1, y= 3; x= 2, y= 5

(iii) 2x+y=6

Sol :

⇒y=6-2x

When x=0 , y=6-2×0=6

When x=1 , y=6-2×1=6-2=4

When x=2 , y=6-2×2=6-4=2

⇒ x= 0, y= 6; x= 1, y= 4; x= 2, y= 2

(iv) 2x-3y=1

Sol :

⇒-3y=1-2x

When x= -1 ,

When x=2 ,

When x=-4 ,

⇒ x= -1, y= -1; x= 2, y= 1; x= -4, y= -3

Question 8

Find four solutions of the equations :

(i) x+2y=6

Sol :

⇒2y=6-x

When x=8 ,

When x=2 ,

When x= -2 ,

When x= 0 ,

⇒ x= 8, y= -1; x= 2, y= 2; x= -2, y= 4; x= 0, y= 3

(ii) x+y=0

Sol :

⇒y=-x

When x=1 , y=-(1)=-1

When x=2 , y=-(2)=-2

When x=3 , y=-(3)=-3

When x=4 , y=-(4)=-4

⇒ x= 1, y= -1; x= 2, y= -2; x= 3, y= -3; x= 4, y= -4

(iii) 2x-3(y-2)=0

Sol :

⇒2x-3y+6=0

⇒-3y=2x-6

When x=0 ,

⇒ x= 0, y= 2; x= -3, y= 0; x= 3, y= 4; x= 6, y= 6

(iv) 2(x-1)+3y=4

Sol :

⇒2x-2+3y=4

⇒3y=4+2-2x

When x=0 ,

When x=3 ,

When x=-3 ,

When x=6 ,

⇒ x= 0, y= 2; x= 3, y= 0; x= -3, y= 4; x= 6, y= -2

(v) x=0

Sol :

⇒ x= 0, y= 1; x= 0, y= 2; x= 0, y= 3; x= 0, y= 4

(vi) y=0

Sol :

⇒0x+y+0=0

⇒y=-0x

⇒y=0x

When x=1 , y=0x=0×1=0

When x=2 , y=0x=0×2=0

When x=3 , y=0x=0×3=0

When x=4 , y=0x=0×4=0

⇒ x= 1, y= 0; x= 2, y= 0; x= 3, y= 0; x= 4, y= 0

Question 9

Find solutions of the form x=a, y=0 and x=0 , y=b of the following pair of linear equations. Do they have any such common solution ?

(i) 5x+3y=15 and 5x+2y=10

Sol :

Given equation : 5x+3y=15

⇒3y=15-5x

When x=0 ,

When x=3 ,

⇒x=0 , y=5 ; x=3 , y=0 ..(i)

Given equation : 5x+2y=10

⇒2y=10-5x

When x=0 ,

When x=2 ,

⇒x=0 , y=5 ; x=2 , y=0 ..(ii)

From (i) and (ii) , we can say that

Yes common solution is x=0 , y=5

(ii) x+y=3 and 2x+5y=12

Sol :

Given equation : x+y=3

⇒y=3-x

When x=0 , y=3-0=3

When x=3 , y=3-3=0

⇒x=0 , y=3 ; x=3 , y=0 ..(i)

Given equation : 2x+5y=12

⇒5y=12-2x

When x=0 ,

When x=6 ,

⇒x=0 , y=2.4 ; x=6 , y=0 ..(i)

From (i) and (ii) , we can say that

No

(iii) 2x+3y=1 and x-y=1

Sol :

Given equation : 2x+3y=1

⇒3y=1-2x

When x=0 ,

When x=1/2 ,

⇒x=0 , y= 1/3 ; x= 1/2 , y=0 ..(i)

Given equation : x-y=1

⇒-y=1-x

⇒y=x-1

When x=0 , y=0-1=-1

When x=1 , y=1-1=0

⇒x=0 , y=-1 ; x=1 , y=0..(ii)

From (i) and (ii) , we can say that

NO

(iv) 3x-9y=6 and 2x-6y=8

Sol :

Given equation : 3x-9y=6

⇒-9y=6-3x

When x=2 ,

When x=0 ,

⇒x=2 , y=0 ; x=0 , y=-2/3..(i)

Given equation :2x-6y=8

⇒-6y=8-2x

When x=0 ,

When x=4 ,

⇒x=0 , y=-4/3 ; x=4 , y=0..(ii)

From (i) and (ii) , we can say that

NO

Type 4

Problems based on graph of a linear equation in two variables.

WORKING RULE:

1. Write y

2.

3.

4.

5.

6.

7.

8.

Question 10

(i) Draw the graph of x+2y=6

Sol :

(ii) Draw the graph of x+y=7

Sol :

(iii) Draw the graph of x-y=7

Sol :

(iv) Draw the graph of 2x+y=3

Sol :

Question 11

Draw the graph of the following equations and find the coordinates of the points where the cuts the axes :

(i) 3x+2y=6

Sol :

(ii) 2x+3y=12

Sol :

(iii) y+3x=9

Sol :

(iv) (x-4)-y+4=0

Sol :

(v) 4x-5y=20

Sol :

(vi) x+y=0

Sol :

Question 12

From the choices given below choose the equation whose graphs have been given:

(a) For figure (i):

(i) x+y=0

(ii) y=2x

(iii) y=x

(iv) y=2x+1

(b) For figure (ii):

(i) x+y=0

(ii) y=2x

(iii) y=2x+4

(iv) y=x-4

(c) For figure(iii):

(i) x+y=0

(ii) y=2x

(iii) y=2x+1

(iv) y=2x-4

Sol :

Question 13

Draw the graph of the following :

(i) x=3

Sol :

(ii) x=-2

Sol :

(iii) y=2

Sol :

(iv) y=-3

Sol :

(v) x-3=0

Sol :

(vi) 2x-3=0

Sol :

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