Question

Write an equation of a line passing through the point representing solution of the pair of linear equations x + y = 2 and 2x - y = 1 How many such lines can we find?

Answer 1

Given pair of linear equations is

x + y - 2 = 0 …(i)

and 2x - y - 1 = 0 …(ii)

On comparing with ax + by + c = 0

Here, a₁ = 1, b₁ = 1, c₁ = - 2;

And a₂ = 2, b₂ = - 1, c₂ = - 1;

a₁ /a₂ = 1/2

b₁ /b₂ = - 1

c₁ /c₂ = 2

since a₁/a₂ ≠ b₁/b₂

So, both lines intersect at a point. Therefore, the pair of equations has a unique solution. Hence, these equations are consistent.

Now, x + y = 2 or y = 2-x

If x = 0 then y = 2 and if x = 2 then y = 0

and

If x = 0 then y = - 1 if x = 1/2  then y = 0 and if x = 1 then y = 1

Plotting the points A (2,0) and B(0,2), we get the straight line AB. Plotting the points C(0,- 1) and D(1/2, 0), we get the straight line CD.

The lines AB and CD intersect at E (1, 1).

Hence, infinite lines can pass through the intersection point of linear equations x + y = 2 and 2x - y - 1 = 0 i.e. E (1, 1) like as y = x, x + 2y = 3, x + y = 2 and so on.