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How many solutions does the system of equations below have?\newline2x−2y−z=32x - 2y - z = 3\newline−x−y+z=−12-x - y + z = -12\newline−3x+y+3z=−12-3x + y + 3z = -12\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

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Q. How many solutions does the system of equations below have?\newline2x−2y−z=32x - 2y - z = 3\newline−x−y+z=−12-x - y + z = -12\newline−3x+y+3z=−12-3x + y + 3z = -12\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Multiply and Add Equations: First, let's multiply the second equation by 22 so we can add it to the first equation and eliminate yy.\newline2(−x−y+z)=2(−12)2(-x - y + z) = 2(-12)\newline−2x−2y+2z=−24-2x - 2y + 2z = -24
  2. Combine Equations: Now, add the new equation to the first one:\newline(2x−2y−z)+(−2x−2y+2z)=3+(−24)(2x - 2y - z) + (-2x - 2y + 2z) = 3 + (-24)\newlineThe xx terms and zz terms cancel out, leaving:\newline−4y=−21-4y = -21

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