Prove that The lines $2 x + y = 2$ , $x + 2 y = 1$ and $x + y = 1$ pass through the same point.

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Solution

We shall first find the point of intersection of the line $2 x + y = 2$ and $(x + 2 y = 1$ ) , as follows:

$2 x + y = 2$ - - - - - (1)

$x + 2 y = 1$ - - - - - (2)

$2 \times (2)$ gives $2 x + 4 y = 2$ - - - - - (3)

(1) - (3) gives: $- 3 y = 0$

$⟹ y = 0$

Substituting $y = 0$ in (1), we get $x = 1$ .

$∴$ Point of intersection is $(1, 0)$ .

Putting this point in the L.H.S. of $x + y = 1$ , we get,

$1 + 0$

= $1$

= R.H.S.

Hence all the given lines pass through the same point.

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