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How many solutions does the system of equations below have?\newline2x−y−z=−132x - y - z = -13\newline3x−2y−z=−203x - 2y - z = -20\newline−3x+y+2z=19-3x + y + 2z = 19\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?\newline2x−y−z=−132x - y - z = -13\newline3x−2y−z=−203x - 2y - z = -20\newline−3x+y+2z=19-3x + y + 2z = 19\newlineChoices:\newline(A)no solution\newline(B)one solution\newline(C)infinitely many solutions
  1. Combine Equations: Combine the first two equations to eliminate zz.(2x−y−z)−(3x−2y−z)=−13−(−20)(2x - y - z) - (3x - 2y - z) = -13 - (-20)−x+y=7-x + y = 7
  2. Add Equations: Now, let's add the first and the third equation to eliminate zz again.(2x−y−z)+(−3x+y+2z)=−13+19(2x - y - z) + (-3x + y + 2z) = -13 + 19−x+z=6-x + z = 6
  3. Solve for y and z: We have two new equations:\newline−x+y=7-x + y = 7\newline−x+z=6-x + z = 6\newlineLet's solve for y and z in terms of x.\newliney=x+7y = x + 7\newlinez=x+6z = x + 6
  4. Substitute Back: Substitute yy and zz back into one of the original equations, let's use the first one.\newline2x−(x+7)−(x+6)=−132x - (x + 7) - (x + 6) = -13\newline2x−x−7−x−6=−132x - x - 7 - x - 6 = -13
  5. Simplify Equation: Simplify the equation. 2x−x−x=−13+7+62x - x - x = -13 + 7 + 6 0x=00x = 0
  6. Infinite Solutions: Since 0x=00x = 0 is true for all xx, we have infinitely many solutions.

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