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How many solutions does the system of equations below have?\newline3x−3y+3z=−183x - 3y + 3z = -18\newlinex−y+z=−6x - y + z = -6\newline−2x+2y−2z=12-2x + 2y - 2z = 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−3y+3z=−183x - 3y + 3z = -18\newlinex−y+z=−6x - y + z = -6\newline−2x+2y−2z=12-2x + 2y - 2z = 12\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Simplify Equations: First, let's simplify the equations by dividing the first and third equations by 33 and −2-2, respectively, to see if they are multiples of the second equation.3x−3y+3z=−183x - 3y + 3z = -18 becomes x−y+z=−6x - y + z = -6 after dividing by 33.−2x+2y−2z=12-2x + 2y - 2z = 12 becomes x−y+z=−6x - y + z = -6 after dividing by −2-2.
  2. Check for Multiples: Now we have:\newline11. x−y+z=−6x - y + z = -6\newline22. x−y+z=−6x - y + z = -6\newline33. x−y+z=−6x - y + z = -6\newlineAll three equations are the same, which means every point on the line of the second equation is also on the lines of the first and third equations.
  3. Identify Same Equations: Since all three equations represent the same line, the system has infinitely many solutions because there are an infinite number of points where these lines intersect (which is the line itself).

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