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Cramer’s Rules :

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☼ Cramer’s Rules :

If l₁ : a₁x + b₁y + c₁ = 0

and l₂ : a₂x + b₂y + c₂ = 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 P ).

This unique solution, P = $\left (\frac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}},- \frac{a_{1}c_{2}-a_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}} \right )$

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

In this case, we get the no solution.

i.e. The Solution Set = $\emptyset$

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

In this case, we get infinite solutions.

i.e. The Solution Set = { ( x, y ) / a₁ x + b₁y + c₁ = 0 : x, y $\in$ R }