TsAmE said:
Under what conditions on a, b and c is the following system of equations consistent?
2x + 4y + z = a
6x + 14y + 2z = b
4x + 10y + z = c
\(\displaystyle .\)
\(\displaystyle .\)
\(\displaystyle .\)
but the correct answer was b = a + c?
\(\displaystyle 2\ \ \ \ \ \ 4 \ \ \ \ 1 \ \ \ \ a\)
\(\displaystyle 6\ \ \ \ 14 \ \ \ \ 2\ \ \ \ b\)
\(\displaystyle 4 \ \ \ \ 10 \ \ \ \ 1 \ \ \ \ c\)
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\(\displaystyle -3R_1 + R_2 \longrightarrow\)
\(\displaystyle -2R_1 + R_3 \longrightarrow\)
\(\displaystyle 2 \ \ \ \ 4 \ \ \ \ \ \ 1 \ \ \ \ a\)
\(\displaystyle 0 \ \ \ \ 2 \ \ -1 \ \ \ \ b - 3a\)
\(\displaystyle 0 \ \ \ \ 2 \ \ -1 \ \ \ \ c - 2a\)
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\(\displaystyle -R_2 + R_3 \longrightarrow\)
\(\displaystyle 2 \ \ \ \ 4 \ \ \ \ \ \ 1 \ \ \ \ a\)
\(\displaystyle 0 \ \ \ \ 2 \ \ -1 \ \ \ \ b - 3a\)
\(\displaystyle 0 \ \ \ \ 0 \ \ \ \ \ \ 0 \ \ \ \ a + c - b\)
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From here, \(\displaystyle a + c - b = 0 \ \ for \ \ the \ \ system \ \ to \ \ be \ \ consistent.\)
\(\displaystyle Therefore, \ \ \boxed{ b = a + c.}\)
