CBSE 2025, Set 1

Question

Let $2x + 5y - 1 = 0$ and $3x + 2y - 7 = 0$ represent the equations of two lines on which ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.

Edvaya · Accepted Answer

The correct answer is Subjective. Rewrite as $2x + 5y = 1$ and $3x + 2y = 7$. In matrix form $AX = B$: $$\begin{bmatrix}2&5\3&2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix} = \begin{bmatrix}1\7\end{bmatrix}$$ $|A| = 4 - 15 = -11 
eq 0$, so a unique solution exists. $$	ext{adj}(A) = \begin{bmatrix}2&-5\-3&2\end{bmatrix}, \qquad A^{-1} = \frac{1}{-11}\begin{bmatrix}2&-5\-3&2\end{bmatrix}$$ $$X = A^{-1}B = \frac{1}{-11}\begin{bmatrix}2&-5\-3&2\end{bmatrix}\begin{bmatrix}1\7\end{bmatrix} = \frac{1}{-11}\begin{bmatrix}2-35\-3+14\end{bmatrix} = \frac{1}{-11}\begin{bmatrix}-33\11\end{bmatrix} = \begin{bmatrix}3\-1\end{bmatrix}$$ The common point is $\boxed{(3,\ -1)}$.

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