First slide

System of linear equations

Question

If $a + b + c \neq 0,$ then system of equations $(b + c) (y + z) - a x = b - c,$ $(c + a) (z + x) - b y = c - a$ , $(a + b) (x + y) - c z = a - b$ has

Moderate

A

a unique solution
B

no solution
C

infinite number of solutions
D

finitely many solutions

Solution

We can write the above system of equations as
$\begin{array}{l} (a + b + c) (y + z) - a (x + y + z) = b - c \\ (a + b + c) (z + x) - b (x + y + z) = c - a, (a + b + c) (x + y) - c (x + y + z) = a - b \end{array}$
Adding the above equations , we obtain
$\begin{array}{l} 2 (a + b + c) (x + y + z) - (a + b + c) (x + y + z) = 0 \\ \Rightarrow (a + b + c) (x + y + z) = 0 \Rightarrow x + y + z = 0 \Rightarrow y + z = - x \\ ∴ (b + c) (- x) - a x = b - c \Rightarrow x = \frac{c - b}{a + b + c} . similarly, y = \frac{a - c}{a + b + c}, z = \frac{b - a}{a + b + c} \end{array}$

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