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Cramer’s Rules

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☼ Cramer’s Rules ( for homogeneous system ):

If l₁ : a₁x + b a₁y = 0

and l₂ : a a₂x + b a₂y = 0

( Where, a_i, b_i, c_i are real numbers and $a_{i}^{2} + b_{i}^{2}$ $\ne$ 0 : i = 1, 2 )

(i) Intersecting Lines :– If $\frac{a_{1}}{a_{2}} \ne \frac{b_{1}}{b_{2}}$ , then lines l₁ and l₂ are intersecting.

In this case, we get the unique solution ( say point O means Origin ).

This unique solution, O = ( 0, 0 ) ( means only zero solution. )

(ii) Identical Lines :– If $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$ , then lines l₁ and l₂ are identical.

In this case, we get infinite solutions.

i.e. The Solution Set = { ( x, y ) / a₁x + b₁y = 0 : x, y $\in$ R }