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How many solutions does the system of equations below have?\newline3x−y−3z=93x - y - 3z = 9\newline−x+y+2z=−11-x + y + 2z = -11\newline−2x+2y+z=−10-2x + 2y + z = -10\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−3z=93x - y - 3z = 9\newline−x+y+2z=−11-x + y + 2z = -11\newline−2x+2y+z=−10-2x + 2y + z = -10\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Write Equations: First, let's write down the system of equations to see what we're working with:\newline3x−y−3z=93x - y - 3z = 9\newline−x+y+2z=−11-x + y + 2z = -11\newline−2x+2y+z=−10-2x + 2y + z = -10
  2. Check Coefficients: We can use the method of elimination or substitution to solve the system, but before that, let's check if the coefficients of the variables are multiples of each other, which might indicate no solution or infinitely many solutions.
  3. Eliminate Variables: Looking at the coefficients of xx, yy, and zz in each equation, they don't seem to be simple multiples of each other, so we can't immediately conclude that there are no solutions or infinitely many solutions.
  4. Add Equations: Let's try to eliminate one variable by adding or subtracting the equations. If we add the first and second equations, we can eliminate yy:(3x−y−3z)+(−x+y+2z)=9−11(3x - y - 3z) + (-x + y + 2z) = 9 - 112x−z=−22x - z = -2
  5. Simplify Second Equation: Now, let's add the second and third equations to eliminate yy again: (−x+y+2z)+(−2x+2y+z)=−11−10(-x + y + 2z) + (-2x + 2y + z) = -11 - 10 −3x+3z=−21-3x + 3z = -21
  6. Identify Mistake: We can simplify the second equation we got by dividing by −3-3:−3x+3z=−21-3x + 3z = -21x−z=7x - z = 7Oops, we made a mistake here. We should have divided all terms by 33, not just the xx term.

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