High School

Solve the following linear system by Gaussian elimination:

1. [tex]x_1 + 4x_2 + 4x_3 = 16[/tex]
2. [tex]-x_1 - 5x_2 + 5x_3 = 19[/tex]
3. [tex]3x_1 + 7x_2 + 6x_3 = 9[/tex]

Answer :

Answer:

The solution to the given linear system is x₁ = 1265.34, x₂ = -571.67, x₃ = -64.67.

Step-by-step explanation:

To solve the linear system using Gaussian elimination, we start with the augmented matrix:

[ 1 4 4 | -16 ]

[-1 -5 5 | -19 ]

[ 3 7 6 | -9 ]

We'll perform row operations to transform the augmented matrix into row-echelon form.

Step 1: Swap rows R1 and R2:

[-1 -5 5 | -19 ]

[ 1 4 4 | -16 ]

[ 3 7 6 | -9 ]

Step 2: Multiply R1 by -1:

[ 1 5 -5 | 19 ]

[ 1 4 4 | -16 ]

[ 3 7 6 | -9 ]

Step 3: Subtract R1 from R2 and R3:

[ 1 5 -5 | 19 ]

[ 0 -1 9 | -35 ]

[ 0 -8 21 | -66 ]

Step 4: Multiply R2 by -1:

[ 1 5 -5 | 19 ]

[ 0 1 -9 | 35 ]

[ 0 -8 21 | -66 ]

Step 5: Add 5 times R2 to R1:

[ 1 0 -50 | 184 ]

[ 0 1 -9 | 35 ]

[ 0 -8 21 | -66 ]

Step 6: Add 8 times R2 to R3:

[ 1 0 -50 | 184 ]

[ 0 1 -9 | 35 ]

[ 0 0 -3 | 194 ]

Step 7: Divide R3 by -3:

[ 1 0 -50 | 184 ]

[ 0 1 -9 | 35 ]

[ 0 0 1 | -64.67 ]

Step 8: Add 50 times R3 to R1 and 9 times R3 to R2:

[ 1 0 0 | 1265.34 ]

[ 0 1 0 | -571.67 ]

[ 0 0 1 | -64.67 ]

The row-echelon form of the augmented matrix is obtained. Now, we can read the solutions of the linear system directly:

x₁ = 1265.34

x₂ = -571.67

x₃ = -64.67

know more about linear system : brainly.com/question/26544018

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