Question 4. Which of the Following Pairs of Linear Equations are Exercise 3.2 Chapter 3 Pair of Linear Equations in

Question 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

Convert the equation in form of a₁x + b₁y + c₁ = 0  and a₂x+ b₂y + c₂ = 0
We get
x + y  -5 = 0, 2 x + 2 y  -10 = 0
Compare the equation with

We get
a1 = 1,            b1       = 1,                 and c1 = -5
a2 = 2             b2        = 2                   and c2 = - 10

So we get

Dependent and consistent  so it will have many solutions .

Convert the equation in form of
a₁x + b₁y + c₁ = 0  and a₂x+ b₂y + c₂ = 0
We get
x – y -8 = , 3x – 3y -16 = 0
Compare the equation with

We get
a1 = 1,  b1     = - 1,              and c1 = - 8
a2 = 3  b2      =  - 3               and c2 = - 16

So we get

So both lines are Inconsistent
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
Compare the equation with

We get
a1 = 2,            b1       =  1,                and c1 = - 6
a2 = 4             b2        =  - 2               and c2 = - 4

 
So we get

So both lines are consistent

Solve the equations graphically
2x + y – 6 = 0,
Subtract 2x and add 6 both side we get
 y  = 6 – 2x
plug x = 1,2 and 3 we get
y         =  6 - 2 * 1      = 4
 y         = 6 -  2 * 2      = 2
y          =  6 -  2 * 3     = 0

 4 x – 2 y – 4 = 0
Divide by 2 we get
2 x – y - 2                  = 0 
Add y both side we get
2 x – 2            = y
or
y          = 2x – 2
plug x  = 1.2 and 3 we get
y          = 2 * 1 – 2      = 0
y          = 2 * 2 – 2      = 2
y          = 2 * 3 – 2      =  4

Answer x = 2 and y = 2

(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Compare the equation with

We get
a1 = 2,            b1       = - 2,              and c1 = - 2
a2 = 2             b2        =  - 2               and c2 = - 5

So we get

So both lines are Inconsistent