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Solve the system of equations by substitution.\newlinez=−1z = -1\newlinex−3y−2z=5x - 3y - 2z = 5\newline3x+y+z=−23x + y + z = -2

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Q. Solve the system of equations by substitution.\newlinez=−1z = -1\newlinex−3y−2z=5x - 3y - 2z = 5\newline3x+y+z=−23x + y + z = -2
  1. Substitute z=−1z = -1: Substitute z=−1z = -1 into the second equation x−3y−2z=5x - 3y - 2z = 5.\newlinex−3y−2(−1)=5x - 3y - 2(-1) = 5\newlinex−3y+2=5x - 3y + 2 = 5
  2. Isolate x−3yx - 3y: Subtract 22 from both sides to isolate x−3yx - 3y.\newlinex−3y=5−2x - 3y = 5 - 2\newlinex−3y=3x - 3y = 3
  3. Substitute z=−1z = -1: Now substitute z=−1z = -1 into the third equation 3x+y+z=−23x + y + z = -2.
    3x+y+(−1)=−23x + y + (-1) = -2
    3x+y−1=−23x + y - 1 = -2
  4. Isolate 3x+y3x + y: Add 11 to both sides to isolate 3x+y3x + y.\newline3x+y=−2+13x + y = -2 + 1\newline3x+y=−13x + y = -1
  5. Solve for x: Now we have two equations with x and y:\newline11) x−3y=3x - 3y = 3\newline22) 3x+y=−13x + y = -1\newlineLet's solve for x from equation 11).\newlinex=3y+3x = 3y + 3
  6. Substitute x=3y+3x = 3y + 3: Substitute x=3y+3x = 3y + 3 into equation 22) 3x+y=−13x + y = -1.\newline3(3y+3)+y=−13(3y + 3) + y = -1\newline9y+9+y=−19y + 9 + y = -1
  7. Combine like terms: Combine like terms. 10y+9=−110y + 9 = -1
  8. Solve for y: Subtract 99 from both sides to solve for yy.\newline10y=−1−910y = -1 - 9\newline10y=−1010y = -10
  9. Find yy: Divide both sides by 1010 to find yy.\newliney=−1010y = \frac{-10}{10}\newliney=−1y = -1
  10. Substitute y=−1y = -1: Now substitute y=−1y = -1 back into x=3y+3x = 3y + 3 to find xx.\newlinex=3(−1)+3x = 3(-1) + 3\newlinex=−3+3x = -3 + 3\newlinex=0x = 0
  11. Final Solution: We have found x=0x = 0, y=−1y = -1, and we were given z=−1z = -1. The solution is (0,−1,−1)(0, -1, -1).

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