Question

Розв'язати систему лiнiйних рівнянь:

5x₁ + 8x₂x3 = 7
x₁ + 2x₂ + 3x3 = 1
(2x1 - 3x2 + 2x3 = 9
D(051)

Answer 1

Write the augmented matrix of the system:

[5 8 3 | 7]

[1 2 3 | 1]

[2 -3 2 | 9]

Use elementary row operations to transform the matrix into row echelon form:

R2 = R2 - (1/5)R1

R3 = R3 - (2/5)R1

[ 5 8 3 | 7]

[ 0 -6/5 9/5| -2/5]

[ 0 -7/5 -4/5| 29/5]

R3 = R3 - (5/6)R2

[ 5 8 3 | 7]

[ 0 -6/5 9/5 | -2/5]

[ 0 0 11/6 | 5/6]

Solve for the variables starting from the bottom of the matrix:

11/6 x3 = 5/6, so x3 = 5/11

-6/5 x2 + 9/5 x3 = -2/5, so x2 = (9/5)x3 - (2/5)(-6/5) = 9/5(5/11) + 12/55 = 87/55

5x1 + 8x2 + 3x3 = 7, so x1 = (7 - 8x2 - 3x3)/5 = (7 - 8(87/55) - 3(5/11))/5 = -34/55

Therefore, the solution to the system of equations is x1 = -34/55, x2 = 87/55, x3 = 5/11.