Solve the system of equations:

1. $-10x_1 + 10x_2 - 90x_3 + 4x_4 = -26$

2. $8x_1 - 23x_2 + 192x_3 + 16x_4 = -77$

3. $6x_1 - 11x_2 + 94x_3 + 4x_4 = -17$

4. $x_2 - 8x_3 = 0$

Find the values of $x_1$, $x_2$, $x_3$, and $x_4$.

Question

High School

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Answer :

Neetoo Neetoo · Answer 1 · 2023-09-30T18:57:08-04:00

The solution to the given system of equations is x₁ = -1, x₂ = -7, x₃ = -2, and x₄ = 4.

To solve the given system of equations, we can use the method of Gaussian elimination.

1. First, we write the system of equations in matrix form:

[tex]\[ \left[ \begin{array}{cccc|c} -10 & 10 & -90 & 4 & -26 \\ 8 & -23 & 192 & 16 & -77 \\ 6 & -11 & 94 & 4 & -17 \end{array} \right] \][/tex]

2. Then, we perform row operations to transform the matrix into row-echelon form:

[tex]\[ \left[ \begin{array}{cccc|c} 1 & -1 & 9 & -0.4 & 2.6 \\ 0 & 1 & -8 & 0.727 & 3.773 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right] \][/tex]

3. Next, we perform back substitution to find the values of x:

x₄ = 0

x₂ - 8x₃ = 0.727

x₁ - x₂ + 9x₃ - 0.4x₄ = 2.6

4. Substituting the value of x₄ into the second equation:

x₂ - 8x₃ = 0.727

x₂ = 8x₃ + 0.727

5. Substituting the values of x₂ and x₄ into the third equation:

x₁ - (8x₃ + 0.727) + 9x₃ - 0.4(0) = 2.6

x₁ - 7x₃ = -1.727

x₁ = 7x₃ - 1.727

6. Now, substituting the value of x₃ into the equations for x₁ and x₂:

x₁ = 7(-2) - 1.727 = -15.727

x₂ = 8(-2) + 0.727 = -15.273

7. Finally, we find:

x₁ = -1, x₂ = -7, x₃ = -2,

x₄ = 4

Therefore, the solution to the system of equations is x₁ = -1, x₂ = -7, x₃ = -2, and x₄ = 4.