Test of Mathematics Solution Subjective 62 - System of Equations

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.

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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.

i

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.