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How many solutions does the system of equations below have?\newline−3x−3y+3z=12-3x - 3y + 3z = 12\newline2x+2y−2z=−82x + 2y - 2z = -8\newlinex+y−z=−4x + y - z = -4\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?\newline−3x−3y+3z=12-3x - 3y + 3z = 12\newline2x+2y−2z=−82x + 2y - 2z = -8\newlinex+y−z=−4x + y - z = -4\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Multiply and Compare Equations: First, let's multiply the third equation by 22 so we can compare it with the second equation.\newline2(x+y−z)=2(−4)2(x + y - z) = 2(-4)\newline2x+2y−2z=−82x + 2y - 2z = -8
  2. Check Equality of Equations: Now, let's compare the second equation with the new equation we got from doubling the third one.\newline2x+2y−2z=−82x + 2y - 2z = -8 (from the second equation)\newline2x+2y−2z=−82x + 2y - 2z = -8 (from doubling the third equation)\newlineThey are the same, which means the second and third equations are actually the same line.
  3. Check for Multiple of Equations: Next, let's check if the first equation is a multiple of the second equation.\newline−3x−3y+3z=12-3x - 3y + 3z = 12 can be divided by −3-3 to get:\newlinex+y−z=−4x + y - z = -4\newlineThis is the same as the third equation, so all three equations represent the same line.
  4. Infinite Solutions Conclusion: Since all three equations represent the same line, the system has infinitely many solutions because the lines coincide.

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