Solve the following three-variable system:

\[2x + 3y + 4z = 9\]
\[-x + 2y - z = 0\]
\[-2x + 4y + z = 3\]

1 ) 2 x + 3 y + 4 z = 9 2 ) − x + 2 y − z = 0 3 ) − 2 x + 4 y + z = 3 =================================== 3 ) − 2 x + 4 y + z = 3 ⇒ z = 2 x − 4 y + 3 1 ) 2 x + 3 y + 4 z = 9 ⇒ 2 x + 3 y + 4 ( 2 x − 4 y + 3 ) = 9 2 x + 3 y + 8 x − 16 y + 12 = 9 10 x − 13 y = − 3 2 ) − x + 2 y − z = 0 ⇒ − x + 2 y − ( 2 x − 4 y + 3 ) = 0 − x + 2 y − 2 x + 4 y − 3 = 0 − 3 x + 6 y = 3 / : ( − 3 ) x − 2 y = − 1 ⇒ x = 2 y − 1
1 ) 10 x − 13 y = − 3 ⇒ 10 ( 2 y − 1 ) − 13 y = − 3 20 y − 10 − 13 y = − 3 7 y = 7 / : 7 ⇒ y = 1 2 ) x = 2 y − 1 ⇒ x = 2 ⋅ 2 − 1 = 1 3 ) z = 2 x − 4 y + 3 ⇒ z = 2 ⋅ 1 − 4 ⋅ 1 + 3 = 2 − 4 + 3 = 1 A n s . x = y = z = 1

Answered by kate200468 | 2024-06-10

The solution to the three-variable system of equations is x = 1 , y = 1 , z = 1 . This was achieved by expressing one variable in terms of others and substituting back into the equations. The final results provide the values for all three variables that satisfy all given equations.
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Answered by kate200468 | 2024-10-01