First slide

System of linear equations

Question

Given a = x/(y - z), b = y/(z - x), and c = z/(x - y), where x, y, and z are not all zero, then the value of ab + bc + ca is

Moderate

A

0
B

1
C

-1
D

none of these

Solution

$\begin{array}{l} a = x / (y - z) \Rightarrow x - ay + az = 0 (1) \\ b = y / (z - x) \Rightarrow bx + y - bz = 0 (2) \\ c = z / (x - y) \Rightarrow - cx + cy + z = 0 (3) \end{array}$
Since x, y, z are not all zero, the above system has a non-trivial solution. So,
$Δ = |\begin{array}{ccc} 1 & - a & a \\ b & 1 & - b \\ - c & c & 1 \end{array}| = 0$
or 1 + ab + bc + ca = 0

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