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Pair of linear equations in two variables

Standard Equations of Linear Equations –  a₁x + b₁y + c₁  = 0

                                                                                          a₂x + b₂y + c₂   = 0

Linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂   = 0  represents a pair of linear equations that can lead to the following situations.

Linear Equations	Comparison of Ratios	Graphical Solution	Algebraic Solution	Consistent/Inconsistent
a1x + b1y + c1  = 0	a1  =  b1    =  c1	Coincident lines	Multiple solutions	consistent solution
a2x + b2y + c2   = 0	a2           b2            c2			(dependent)
a1x + b1x + c1  = 0	a1  =  b1      c1	Parallel lines	No solution	inconsistent solution
a2x + b2x + c2   = 0	a2           b2            c2
a1x + b1x + c1  = 0	a1    b1      c1	Intersecting lines	Unique (one) solution	consistent solution
a2x + b2x + c2   = 0	a2           b2            c2

Exercise  – 3.1

Q. 1. Form the pair of linear equations for 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.

1.  Total students = 10

            Let the number of girls = x

            Let the number of boys = y

According to the question,

             x  =  y + 4 _ _ _ 1

             x + y  =  10 _ _ _ 2

From equation 1,

x = y + 4                 = Let y = 1                = Let y = 2

Let [y =  0]               x = 1 + 4                    x = 2 + 4

x = 0 + 4                     [ x = 5 ]                     [ x = 6 ]

x	4	5	6
y	0	1	2

From equation 2,

x + y =10                             = Let y = 1                      = Let y = 2

Let  y = 10                     x + 1 = 10                   x + 2 = 10

x + 0  = 10                      x = 10 – 1                    x = 10 – 2

[ x = 10 ]                                [ x  = 9 ]                              [ x  =  8 ]

x	10	9	8
y	0	1	2

X = 7     ,        y = 3

Number of girls = 7

Number of boys = 3

(ii) The total cost of 5 pencils and 7 pens is ₹50, while the total cost of 7 pencils and 5 pens is ₹46. Find the cost of one pencil and the cost of one pen.

Let the price of 1 pencil = Rs. x

    Let the price of 1 pen = Rs. y

According to the question,

      5x + 7y  =  50 _ _ _ _ 1

      7x  + 5y =  46 _ _ _ _  2

From equation 1,

5x + 7y = 50

5x = 50 – 7y

x	10	3	-4
y	0	5	10

From Equation 2,

7x + 5y = 46

5y = 46 – 7x

x	0	5	10
y	9.2	2.2	-4.8

Price of a pencil ( x ) = ₹3

Price of a pen ( y ) = ₹5

Q. 2. By comparing the ratios, determine whether the lines represented by the following pair of equations intersect at a point, are parallel, or are coincident:

1.  5x – 4y + 8 = 0

      7x + 6y – 9 = 0

a₁x + b₁y + c₁  = 0

a₂x + b₂y + c₂   = 0

a₁  = 5  ,    b₁  = -4     ,   c₁  = 8

a₂   = 7  ,   b₂   =  6     ,   c₂   = -9

So these are intersecting lines.

2. 9x + 3y + 12 = 0

   18x + 6y + 24 = 0

a₁  = 9     ,   b₁  =  3     ,   c₁  = 12

a₂   = 18  ,   b₂   =  6     ,   c₂   = 24

So these are coincident lines.

3. 6x – 3y + 10 = 0

    2x  – y   + 9   = 0   

a₁  = 6     ,   b₁  =  – 3     ,   c₁  = 10

a₂   = 2     ,   b₂   =  – 1    ,   c₂   = 9

So these are parallel lines.

Q. 3. By comparing the ratios, determine whether the following pairs of linear equations are consistent or inconsistent.

1. 3x + 2y = 5 ;    2x – 3y = 7

3x + 2y – 5 = 0 ;    2x – 3y – 7 = 0

a₁x + b₁y + c₁ = 0 ;   a₂x + b₂y + c₂   = 0

(Consistent solution)

2. 2x – 3y = 8  ;   4x – 6y = 9

2x – 3y – 8 = 0  ;  4x – 6y – x  = 0

a₁x + b₁x + c₁ = 0 ;  a₂x + b₂x + c₂   = 0

a₁  = 2    ,   b₁  =  – 3     ,   c₁  = – 8

a₂   = 4    ,   b₂   =  – 6    ,   c₂   = – 9

(Inconsistent solution)

4. 5x – 3y = 11 ;  -10x + 6y = – 22

5x – 3y – 11 = 0  ;  -10x + 6y + 22 = 0

a₁x + b₁y + c₁ = 0 ;  a₂x + b₂y + c₂   = 0

a₁  = 5    ,   b₁  =   3    ,   c₁  = 11

a₂   = 10  ,   b₂   =  6    ,   c₂   = 22

Dependent (consistent) solution

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

1. X + y = 5  ,  2x + 2y = 10

X + y – 5 = 0 ,  2x + 2y – 10 = 0

a₁  = 1    ,   b₁  =   1   ,   c₁  = – 5

a₂   = 2    ,   b₂   =  2    ,   c₂   = – 10

Dependent (consistent) solution

2. x – y = 8  ,  3x – 3y = 16

X – y – 8 = 0 ,   3x – 3y – 16 = 0

a₁  = 1    ,   b₁  =  -1   ,   c₁  = – 8

a₂   = 3    ,   b₂   =  3    ,   c₂   = – 16

(Inconsistent solution)

3. 2x + y – 6 = 0  ,  4x – 2y – 4 = 0

a₁  = 2    ,   b₁  =  1   ,   c₁  = – 6

a₂   = 4    ,   b₂   = -2  ,   c₂   = – 4

(Consistent solution)

4. 2x – 2y – 2  ,  4x – 4y – 5 = 0

a₁  = 2    ,   b₁  =  -2   ,   c₁  = – 2

a₂   = 4    ,   b₂   =  -4  ,   c₂   = – 5

(Inconsistent solution)

Q. 5. The semi-perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

5. Let the length of the rectangular garden = x

    Let the width of the rectangular garden = y

According to the question,

x = y + 4  -(1)

Semiperimeter = 36 m

x + y = 36

From Equation 1,

x = y + 4

Let y = 0

X = 0 + 4

X = 4

Let y = 1

X = 1 + 4

X = 5

Let y = 2

X = 2 + 4

X = 6

x	4	5	6
y	0	1	2

From Equation 2,,

X + y = 36

X = 36 – y

Let y = 10

X = 36 – 10

X = 26

Let y = 12

X = 36 – 12

X = 24

Let y = 14

X = 36 – 14

X = 22

x	26	24	22
y	10	12	14

X = 20

Y  = 16