Question

The equations x+2y=7 and ax+ay=b have no solutions if​

Answer 1

Complete Question :- The Equations x+2y=7 and ax+ay=b have no solution if :-

a) a = 4, b = 14

b) a = 0, b = 14

c) a = 2, b = 7

d) a = 4, b = 9

Concept used :-

• If we consider two linear equations in two variables, say :-

→ a1x + b1y + c1 = 0

→ a2x + b2y + c2 = 0

Then :-

→ Both the straight lines will be parallel if :-

• a1/a2 = b1/b2 ≠ c1/c2.
• In this case , the system will have no solution.

Solution :-

given two equations,

→ x + 2y - 7 = 0

→ ax + ay - b = 0

comparing them with equations,

→ a1x + b1y + c1 = 0

→ a2x + b2y + c2 = 0

we get,

• a1 = 1, b1 = 2, c1 = (-7)
• a2 = a , b2 = a , c2 = (-b)

so, for no solutions,

→ a1/a2 = b1/b2 ≠ c1/c2.

→ 1/a = 2/a ≠ (-7)/(-b)

→ 1/a = 2/a ≠ 7/b

checking options now,

a) a = 4, b = 14

→ 1/4 = 2/4 ≠ 7/14

→ 1/4 ≠ 1/2 = 1/2

Not satisfy .

b) a = 0, b = 14

→ 1/0 = 2/0 ≠ (7/14)

→ undefined = undefined ≠ 1/2

Satisfy .

c) a = 2, b = 7

→ 1/2 = 2/2 ≠ 7/7

→ 1/2 ≠ 1 = 1

Not satisfy .

d) a = 4, b = 9

→ 1/4 = 2/4 ≠ 7/9

→ 1/4 ≠ 1/2 ≠ 7/9

Not satisfy .

As we can see that, only option (b) satisfy the given condition for no solutions .

therefore, in order for given equations have no solutions (b) a = 0, b = 14 .

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Answer 2

Given :  The equations x+2y=7 and ax+ay=b have no solutions  

To Find : Condition

Solution:

Pair of linear equations

a₁x  +  b₁y + c₁  =  0

a₂x  +  b₂y + c₂  =  0

Consistent

if a₁/a₂ ≠ b₁/b₂   (unique solution  and lines intersects each others)

  a₁/a₂ = b₁/b₂ = c₁/c₂   (infinite solutions and line coincide each other )

Inconsistent

if  a₁/a₂ = b₁/b₂ ≠  c₁/c₂  ( No solution , lines are parallel to each other)

x+2y=7

ax+ay=b

will have no solution if  

1/a   =  2/a   ≠  7/b

1/a   =  2/a

is not possible  for any value of  a    

Hence  there is no such condition where equation does not have no solution

as 1/a   ≠ 2/a  Hence unique solution   for all value of a and b .

Understanding it in different way

linear Equation have no solution when they have same slope but different intercepts

x+2y=7  =>   y = -x/2 + 7/2  Hence slope = -1/2

ax + ay = b => y  = -x  + b/a   Hence slope = - 1

As slope of lines are different irrespective of the values of a and b

Hence lines would be intersecting and will have unique solution

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