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How many solutions does the system of equations below have?\newline3x+y+2z=−153x + y + 2z = -15\newline−3x−y−2z=15-3x - y - 2z = 15\newline−2x−2y+z=12-2x - 2y + z = 12\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions

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Determine the number of solutions to a system of equations in three variablesright chevron icon

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Q. How many solutions does the system of equations below have?\newline3x+y+2z=−153x + y + 2z = -15\newline−3x−y−2z=15-3x - y - 2z = 15\newline−2x−2y+z=12-2x - 2y + z = 12\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Addition of Equations: Add the first two equations to see if they cancel each other out.\newline3x+y+2z=−153x + y + 2z = -15\newline−3x−y−2z=15-3x - y - 2z = 15\newline-----------------\newline0=00 = 0
  2. Verification of Equations: Since 0=00 = 0 is a true statement, it means the first two equations are multiples of each other and represent the same plane.
  3. Checking Third Equation: Now, check if the third equation is also a multiple of the first one.\newline−2x−2y+z=12-2x - 2y + z = 12\newlineIf we multiply the first equation by −2/3-2/3, we should get the third equation if they are the same plane.\newline(−2/3)(3x+y+2z)=(−2/3)(−15)(-2/3)(3x + y + 2z) = (-2/3)(-15)\newline−2x−(2/3)y−(4/3)z=10-2x - (2/3)y - (4/3)z = 10\newlineThis is not the same as the third equation, so there is a mistake here.

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