**Exercise 3.2**

**1.Form the pair of linear equations in the following problems, and find •their solution graphically.**

#### Hence, the cost of one pencil is Rs 3 and that of one pen is Rs 5.

**Verification**: Put x = 3 and y = 5 in (1) and (2), we find that both the equations are satisfied.

**2.On comparing the ratios** _{}

**find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.**

*(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

*(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

**Sol**. *(i) *The given pair of linear equations are

5x- 4y +8= 0 and, …(1)

#### and, 7x +6y – 9 = 0…(2)

Here, _{}

#### => (1) and (2) are intersecting lines.

*(ii)* The given pair of linear equations are

9x +3y +12 = 0

#### and, 18x +6y +24 = 0

Here, _{}

#### => (1) and (2) are coincident lines.

*(iii) *The given pair of linear equations are

#### 6r – 3y + 10 = 0 …(1)

and, x2 – y + 9 = 0 ….(2)

#### => (1) and (2) are parallel lines.

**3.Oncomparing the ratios** _{}

**find out whether the following pairs of linear equations are consistent, or inconsistent.**

*(i)* 3x+2y=5;2x-3y=7

*(i)*3x+2y=5;2x-3y=7

*(ii) *2x – 3y= 8; 4x – 6y= 9

*(ii)*2x – 3y= 8; 4x – 6y= 9

*(iii)* _{}**= 7;9x – 10y = 14**

*(iii)*

*(iv)* 5x-3y =11;- 10x+6y=-22

**(v)** _{}**+2y=8;28+34=12.**

*(iv)*5x-3y =11;- 10x+6y=-22

**(v)**

**Sol. ***(i)* 3x + 2y = 5; 2x – 3y = 7

a_{1} = 3, b_{1}= 2, c_{1}, = 5; a_{2}= 2, b_{2} = – 3, C_{2} = 7

#### =>The pair of linear equations si consistent.

*(ii) *2x – 3y = 8; 4x – 6y = 9

#### a_{1} = 2, b_{1}= -3, c_{1}, = 8; a_{2}= 4, b_{2} = – 6, C_{2} = 9

#### => The pair of linear equations si inconsistent.

#### => The pair of linear equations is consistent.

#### (iv)5x- 3y =1; – 10x +6y=- 22

a_{1} = 5, b_{1}= -3, c_{1}, =11; a_{2}= -10, b_{2} = 6, C_{2} =-22

**=> The pair of linear equation is consistent.**

#### (v) _{} +2y=8:2x+3y =12

_{} b_{1}= 2, c_{1}, =8; a_{2}= 2, b_{2} = 3, C_{2} =12

**4. Which of the following pairs of linear equations are consistent ? Obtain the solution in such cases graphically.**

*(i)* x+y=5, 2x+2y=10

*(ii)* x – y = 8, 3x-3y = 16

*(iii)* 2x+3-6=0, 47-23-4=0

*(iv)* 2x-2y-2=0, 4x-4y-5=0

*(i)*x+y=5, 2x+2y=10

*(ii)*x – y = 8, 3x-3y = 16

*(iii)*2x+3-6=0, 47-23-4=0

*(iv)*2x-2y-2=0, 4x-4y-5=0

**Sol.** (i) Graph of x +y= 5 :

Wehave,x*+y=5 = > y=5-x *

#### When x=0, y=5; when x=5, y=0

#### Thus, we have the following table:* *

#### Plotting the points A(0, 5) and B(5, 0) on the graph paper. Join A and B and extend it on both sides to obtain the graph of x +y=5.

#### Graph of 2x + 2y = 10 :

#### We have 2x+2y=10 => 2y=10-2x

#### => y=5-x

#### When x=1,y=5- 1=4; when x=2,y=5- 2=3

Thus, we have the following table:

#### Plotting the points C(1, 4) and D(2, 3) on the graph paper and drawing a line passing through these points on the same graph paper, we obtain the graph of 2x +2y = 10. We find that C and db othlie on the graph of x + y = 5.

#### Thus, the graphs of the two equations are coincident. Consequently, every solution of one equation is a solution of the other.

Hence, the system of equations has infinitely many solutions, i.e., **consistent**.

*(ii)* Graph of x – y = 8 :

We have,x-y=8 = > y=x-8

When x = 0, y = – 8; when x = 8, y= 0

Thus, we have the following table

#### Plotting the points A(0, – 8) and B(8, 0) on a graph paper. Join A and B and extend it on both sides to obtain the graph of x- y = 8 as shown.

Graph of 3x – 3y = 16 :

Wehave, 3x-3y=16 => 3y=3x-16 => y=_{}

#### Thus, we have the following table:

#### graph paper. Join C and D and extend it on both sides to obtain the graph of 3x- 3y = 16 as shown.

We find the graphs of x- y=8and 3x – 3y= 16 are parallel. So, the two lines have no common point. Hence, the given equations has no solution, i.e., **inconsistent.**

(iii) Graph of 2x +y- 6 = 0 :

#### Wehave,2x+y-6=0 => y=6-2x

When x = 0, y = 6- 0 = 6; when x = 3, y = 6 – 6 = 0

Thus, we have the following table:

#### Plot the points A(0, 6) and B(3, 0) on a graph paper. Join A and B and extend it on both sides to obtain the graph of 2x +y – 6 = 0 as shown.

Graph of 4x- 2y – 4 =0 :

We have,

4x- 2y 4-=0 => 2y=4x-4>= > y=2x- 2

#### When x = 0, y = – 2; when x = 1, y = 0

Thus, we have the following table :

## Plotting the points C(0, -2) and D(2, 0) on the same graph and drawing a line joining them

asP(2, )2 shown.

#### Clearly, the two lines intersect at point P(2, 2).

#### Hence, x= 2, y = 2 is the solution of the given equations, i.e., **consistent**.

**Verification:** Putting x = 2, y = 2 in the given equations, we find that both the equations are satisfied.

#### (iv) Graph of 2x – 2y – 2 =0 :

We have,

2x- 2y-2=0 => 2y=2x-2 = > y=x-2

#### When x=2,y=0; whenx=0,y=- 2

#### Thus, we have the following table:

#### Plot the points A(2, 0) and B(0, – 2) on a graph paper. Join A and B and extend it on both sides to obtain thegraphof 2x-2y- 2=0.

Graph of 4x – 4y – 5 =0 :

We have,

4x-4y-5=0 => 4y=5 => 5-4x => y= _{}

_{}

#### Thus, we have the following table:

#### graph paper. Join C and D and extend it on both sides to obtain the graph of 4x – 4y – 5 =0 as shown.

#### We find the graphs of these equations are parallel lines. So, the two lines have no common point. Hence, the given system of equations has no solution, i.e., **inconsistent.**

**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.**

**Sol.**Let the length of the garden be x m and its width be y m.

#### Hence, Length = 11 m and width = 7 m.

**6.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

Sol. Given linear equation is 2x + 3y – 8 = 0

(i) intersecting lines (ii) parallel lines (iii) coincident lines

Sol

#### (i) For intersecting lines, we know that

#### Any line intersecting line may be taken as

#### 5x+2y- 9=0

#### (ii) For parallel lines, _{}

. Any line parallel to (1) may be taken as

_{}