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This is a Test of Mathematics Solution Subjective 62 (from ISI Entrance).

The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## Problem

Consider the system of equations x + y = 2, ax + y = b. Find conditions on a and b under which
(i) the system has exactly one solution;
(ii) the system has no solution;
(iii) the system has more than one solution.

## Solution

### Key Idea

Solution to the linear equations

$$a_{11} x + a_{12} y = b_1 \\ a_{21} x + a_{22} y = b_2$$

$$a_{11} \times a_{22} – a_{12} \times a_{21} \neq 0$$ implies there is a unique solution.

$$a_{11} \times a_{22} – a_{12} \times a_{21} = 0$$ implies there is either no solution or infinitely many solution.

No solution if: $$\frac{a_{11}}{a_{21} } = \frac{a_{12}}{a_{22}} \neq = \frac{b_1}{b_2}$$; infinitely many solutions other wise.

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