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43{:[x_(1)+x_(2)+x_(3)=-2],[3x_(1)-4x_(3)=22],[x_(1)+2x_(2)+5x_(3)=-18]:}

43x1+x2+x3=−23x1−4x3=22x1+2x2+5x3=−18 43 \begin{aligned} x_{1}+x_{2}+x_{3} & =-2 \\ 3 x_{1}-4 x_{3} & =22 \\ x_{1}+2 x_{2}+5 x_{3} & =-18\end{aligned}

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Q. 43x1+x2+x3=−23x1−4x3=22x1+2x2+5x3=−18 43 \begin{aligned} x_{1}+x_{2}+x_{3} & =-2 \\ 3 x_{1}-4 x_{3} & =22 \\ x_{1}+2 x_{2}+5 x_{3} & =-18\end{aligned}
  1. Identify Equations: Identify the system of equations:\newline11. x1+x2+x3=−2x_1 + x_2 + x_3 = -2\newline22. 3x1−4x3=223x_1 - 4x_3 = 22\newline33. x1+2x2+5x3=−18x_1 + 2x_2 + 5x_3 = -18
  2. Express x11: Use the first equation to express x1x_1 in terms of x2x_2 and x3x_3:\newlinex1=−2−x2−x3x_1 = -2 - x_2 - x_3\newlineSubstitute this expression for x1x_1 in the other equations.
  3. Substitute in 22nd Eq: Substitute x1x_1 in the second equation:\newline3(−2−x2−x3)−4x3=223(-2 - x_2 - x_3) - 4x_3 = 22\newline−6−3x2−3x3−4x3=22-6 - 3x_2 - 3x_3 - 4x_3 = 22\newline−6−3x2−7x3=22-6 - 3x_2 - 7x_3 = 22\newline3x2+7x3=−283x_2 + 7x_3 = -28
  4. Substitute in 33rd Eq: Substitute x1x_1 in the third equation:\newline(−2−x2−x3)+2x2+5x3=−18(-2 - x_2 - x_3) + 2x_2 + 5x_3 = -18\newline−2+x2+4x3=−18-2 + x_2 + 4x_3 = -18\newlinex2+4x3=−16x_2 + 4x_3 = -16
  5. Solve the System: Now solve the system:\newlineFrom 3x2+7x3=−283x_2 + 7x_3 = -28 and x2+4x3=−16x_2 + 4x_3 = -16, eliminate x2x_2:\newlineMultiply the second equation by 33:\newline3x2+12x3=−483x_2 + 12x_3 = -48\newlineSubtract this from the first modified equation:\newline(3x2+7x3)−(3x2+12x3)=−28+48(3x_2 + 7x_3) - (3x_2 + 12x_3) = -28 + 48\newline−5x3=20-5x_3 = 20\newlinex3=−4x_3 = -4
  6. Find x22: Find x2x_2 using x2+4x3=−16x_2 + 4x_3 = -16:\newlinex2+4(−4)=−16x_2 + 4(-4) = -16\newlinex2−16=−16x_2 - 16 = -16\newlinex2=0x_2 = 0
  7. Find x11: Find x1x_1 using x1=−2−x2−x3x_1 = -2 - x_2 - x_3:\newlinex1=−2−0+4x_1 = -2 - 0 + 4\newlinex1=2x_1 = 2

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