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How many solutions does the system of equations below have?\newline3x+y−z=−53x + y - z = -5\newline−2x−2y+z=0-2x - 2y + z = 0\newline3x+2y−3z=83x + 2y - 3z = 8\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?\newline3x+y−z=−53x + y - z = -5\newline−2x−2y+z=0-2x - 2y + z = 0\newline3x+2y−3z=83x + 2y - 3z = 8\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Solve using elimination or substitution: First, let's try to solve the system using elimination or substitution.
  2. Multiply second equation: We can multiply the second equation by 1.51.5 to make the coefficients of zz cancel out when we add it to the third equation.\newline1.5(−2x−2y+z)=1.5(0)1.5(-2x - 2y + z) = 1.5(0)\newlineThis gives us −3x−3y+1.5z=0-3x - 3y + 1.5z = 0
  3. Add new equation to third: Now, add this new equation to the third equation:\newline(3x+2y−3z)+(−3x−3y+1.5z)=8+0(3x + 2y - 3z) + (-3x - 3y + 1.5z) = 8 + 0\newlineThis simplifies to −y−1.5z=8-y - 1.5z = 8
  4. Correct previous mistake: Oops, I made a mistake in the previous step. I should have subtracted 1.5z1.5z, not added it. Let's correct that.\newline(3x+2y−3z)+(−3x−3y+1.5z)=8+0(3x + 2y - 3z) + (-3x - 3y + 1.5z) = 8 + 0\newlineThis simplifies to −y−4.5z=8-y - 4.5z = 8

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