Answer to Question #178268 in Linear Algebra for sarang

Question #178268

Find the conditions on a, b, and c so that the following linear system of equations have a solution.

−2x + y + z = a

x − 2y + z = b

x + y − 3z = c


1
Expert's answer
2021-04-15T07:41:05-0400

"\\Delta=-2\\cdot(6-1)-1\\cdot(-3-1)+1\\cdot(1+2)=-10+4+3=-3\u22600,"


"\\Delta_1=a\\cdot(6-5)-1\\cdot(-3b-c)+1\\cdot(b+2c)=a+4b+3c,"

"\\Delta_2=-2\\cdot(-3b-c)-a\\cdot(-3-1)+1\\cdot(c-b)=4a+5b+3c,"

"\\Delta_3=-2\\cdot(-2c-b)-1\\cdot(c-b)+a\\cdot(1+2)=3a+3b+3c,"


"x=\\frac{\\Delta_1}{\\Delta}=-\\frac a3-\\frac 43 b-c,"

"y=\\frac{\\Delta_2}{\\Delta}=-\\frac 43 a-\\frac 53 b-c,"

"z=\\frac{\\Delta_3}{\\Delta}=-a-b-c,"


there is a solution for any a, b, c.


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