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Solve the system of equations by elimination.\newlinex+2y−2z=−18x + 2y - 2z = -18\newline−3x+2y+2z=−2-3x + 2y + 2z = -2\newline−x+3y−2z=−20-x + 3y - 2z = -20

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Q. Solve the system of equations by elimination.\newlinex+2y−2z=−18x + 2y - 2z = -18\newline−3x+2y+2z=−2-3x + 2y + 2z = -2\newline−x+3y−2z=−20-x + 3y - 2z = -20
  1. Eliminate z by adding equations: Add the first and second equations to eliminate z.\newline(x+2y−2z)+(−3x+2y+2z)=−18+(−2)(x + 2y - 2z) + (-3x + 2y + 2z) = -18 + (-2)\newline−2x+4y=−20-2x + 4y = -20\newlineDivide by −2-2 to simplify.\newlinex−2y=10x - 2y = 10
  2. Find yy by dividing: Now, add the first and third equations to eliminate zz.(x+2y−2z)+(−x+3y−2z)=−18+(−20)(x + 2y - 2z) + (-x + 3y - 2z) = -18 + (-20)3y=−383y = -38Divide by 33 to find yy.y=−383y = \frac{-38}{3}
  3. Substitute to find xx: Substitute yy into x−2y=10x - 2y = 10 to find xx.
    x−2(−383)=10x - 2(-\frac{38}{3}) = 10
    x+763=10x + \frac{76}{3} = 10
    Multiply by 33 to clear the fraction.
    3x+76=303x + 76 = 30
    Subtract 7676 from both sides.
    3x=−463x = -46
    Divide by 33 to find xx.
    yy22
  4. Find z by substitution: Substitute xx and yy into the first equation to find zz.
    (−46/3)+2(−38/3)−2z=−18(-46 / 3) + 2(-38 / 3) - 2z = -18
    Multiply by 33 to clear fractions.
    −46+2(−38)−6z=−54-46 + 2(-38) - 6z = -54
    −46−76−6z=−54-46 - 76 - 6z = -54
    Combine like terms.
    −122−6z=−54-122 - 6z = -54
    Add 122122 to both sides.
    −6z=68-6z = 68
    Divide by yy00 to find zz.
    yy22

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