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Given the system of equations:\newline2x−y−3z=−12 2x - y - 3z = -12 \newline3x+2y−2z=−20 3x + 2y - 2z = -20 \newline−x−2y+z=19 -x - 2y + z = 19

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Q. Given the system of equations:\newline2x−y−3z=−12 2x - y - 3z = -12 \newline3x+2y−2z=−20 3x + 2y - 2z = -20 \newline−x−2y+z=19 -x - 2y + z = 19
  1. Label Equations: First, let's label the equations for easy reference:
    11) 2x−y−3z=−122x - y - 3z = -12
    22) 3x+2y−2z=−203x + 2y - 2z = -20
    33) −x−2y+z=19-x - 2y + z = 19
  2. Eliminate y: Let's eliminate yy by adding equation 11 and equation 33:\newline(2x−y−3z)+(−x−2y+z)=−12+19(2x - y - 3z) + (-x - 2y + z) = -12 + 19\newline2x−y−3z−x−2y+z=72x - y - 3z - x - 2y + z = 7\newlinex−3y−2z=7x - 3y - 2z = 7
  3. Eliminate y: Now, let's eliminate yy by adding equation 22 and equation 33: \newline(3x+2y−2z)+(−x−2y+z)=−20+19(3x + 2y - 2z) + (-x - 2y + z) = -20 + 19 \newline3x+2y−2z−x−2y+z=−13x + 2y - 2z - x - 2y + z = -1 \newline2x−z=−12x - z = -1
  4. New Equations: We now have two new equations:
    44) x−3y−2z=7 x - 3y - 2z = 7
    55) 2x−z=−1 2x - z = -1
  5. Solve for zz: Solve equation 55 for zz:\newline2x−z=−12x - z = -1\newlinez=2x+1z = 2x + 1
  6. Substitute z z into equation: Substitute z=2x+1 z = 2x + 1 into equation 4 4 :
    x−3y−2(2x+1)=7 x - 3y - 2(2x + 1) = 7
    x−3y−4x−2=7 x - 3y - 4x - 2 = 7
    −3x−3y−2=7 -3x - 3y - 2 = 7
    −3x−3y=9 -3x - 3y = 9
    x+y=−3 x + y = -3
  7. Solve for y y : Solve x+y=−3 x + y = -3 for y y : y=−3−x y = -3 - x
  8. Substitute yy and zz into equation: Substitute y=−3−xy = -3 - x and z=2x+1z = 2x + 1 into equation 11: 2x−(−3−x)−3(2x+1)=−122x - (-3 - x) - 3(2x + 1) = -12 2x+3+x−6x−3=−122x + 3 + x - 6x - 3 = -12 −3x=−12-3x = -12 x=4x = 4
  9. Substitute xx into yy: Substitute x=4x = 4 into y=−3−xy = -3 - x:\newliney=−3−4y = -3 - 4\newliney=−7y = -7
  10. Substitute xx into zz: Substitute x=4x = 4 into z=2x+1z = 2x + 1:\newlinez=2(4)+1z = 2(4) + 1\newlinez=9z = 9

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