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How many solutions does the system of equations below have?\newline3x−3y+3z=33x - 3y + 3z = 3\newline−x+y−z=−1-x + y - z = -1\newline−2x+2y−2z=−2-2x + 2y - 2z = -2\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=33x - 3y + 3z = 3\newline−x+y−z=−1-x + y - z = -1\newline−2x+2y−2z=−2-2x + 2y - 2z = -2\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Simplify first equation: Divide the first equation by 33 to simplify.(3x−3y+3z)/3=3/3(3x - 3y + 3z) / 3 = 3 / 3x−y+z=1x - y + z = 1
  2. Simplify second equation: Notice that the second equation is already simplified. −x+y−z=−1-x + y - z = -1
  3. Simplify third equation: Divide the third equation by −2-2 to simplify.−2x+2y−2z−2=−2−2\frac{-2x + 2y - 2z}{-2} = \frac{-2}{-2}x−y+z=1x - y + z = 1
  4. Compare simplified equations: Compare the simplified equations.\newlineFirst equation: x−y+z=1x - y + z = 1\newlineSecond equation: −x+y−z=−1-x + y - z = -1\newlineThird equation: x−y+z=1x - y + z = 1
  5. Eliminate variables: Add the first and second equations to eliminate variables.\newline(x−y+z)+(−x+y−z)=1+(−1)(x - y + z) + (-x + y - z) = 1 + (-1)\newline0=00 = 0
  6. Check for dependency: The addition of the first and second equations results in a true statement, indicating that the equations are dependent.
  7. Identify same plane: Since the third equation is identical to the first, all three equations represent the same plane in three-dimensional space.
  8. Infinite solutions: The system of equations has infinitely many solutions because all three equations describe the same plane.

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