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How many solutions does the system of equations below have?\newline−3x+3y+z=−6-3x + 3y + z = -6\newline3x−3y−z=63x - 3y - z = 6\newline−2x+2y−3z=11-2x + 2y - 3z = 11\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?\newline−3x+3y+z=−6-3x + 3y + z = -6\newline3x−3y−z=63x - 3y - z = 6\newline−2x+2y−3z=11-2x + 2y - 3z = 11\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Add Equations: Add the first two equations to eliminate xx, yy, and zz.(−3x+3y+z)+(3x−3y−z)=−6+6(-3x + 3y + z) + (3x - 3y - z) = -6 + 60=00 = 0
  2. Check Same Plane: Since 0=00 = 0 is a true statement, it means the first two equations represent the same plane. Now, let's check if the third equation is also the same plane.
  3. Multiply Third Equation: Multiply the third equation by 1.51.5 to compare it with the sum of the first two equations.\newline1.5(−2x+2y−3z)=1.5(11)1.5(-2x + 2y - 3z) = 1.5(11)\newline−3x+3y−4.5z=16.5-3x + 3y - 4.5z = 16.5

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