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How many solutions does the system of equations below have?\newline−3x−y+z=−16-3x - y + z = -16\newline2x+y+z=52x + y + z = 5\newline−x−3y+2z=10-x - 3y + 2z = 10\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−y+z=−16-3x - y + z = -16\newline2x+y+z=52x + y + z = 5\newline−x−3y+2z=10-x - 3y + 2z = 10\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Eliminate y by adding equations: First, let's add the first and second equations to eliminate y.\newline(−3x−y+z)+(2x+y+z)=−16+5(-3x - y + z) + (2x + y + z) = -16 + 5\newlineThis simplifies to:\newline−x+2z=−11-x + 2z = -11
  2. Eliminate yy again by adding equations: Now, let's add the second and third equations to eliminate yy again.$2x+y+z\$2x + y + z + (-x - 33y + 22z) = 55 + 1010\)This simplifies to:x−2y+3z=15x - 2y + 3z = 15
  3. Eliminate x by multiplying and adding equations: We can multiply the first equation by 33 and add it to the third equation to eliminate x.\newline3(−3x−y+z)+(−x−3y+2z)=3(−16)+103(-3x - y + z) + (-x - 3y + 2z) = 3(-16) + 10\newlineThis simplifies to:\newline−10x−6y+5z=−38-10x - 6y + 5z = -38\newlineOops, I made a mistake here. The correct calculation should be:\newline−9x−3y+3z−x−3y+2z=−48+10-9x - 3y + 3z - x - 3y + 2z = -48 + 10\newlineWhich simplifies to:\newline−10x−6y+5z=−38-10x - 6y + 5z = -38

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