Question

Solve the following system of linear equations using the Gauss elimination method
with partial pivoting:
2x1 −x2 +x3 = 4
3x1 +2x2 −4x3 = 1
x1 +4x2 −2x3 = 2

Answer 1

Solution. Let's write the system as

Swap the second and first rows

Multiply the first row by -2/3 and add to the second row; multiply the first row by -1/3 and add to the third row get

Swap the second and third rows

Multiply the second row by 7/10 and add to the third row

Find the roots of the equation

$$x_3=\frac {9}{2}\times \frac{5}{16}=\frac{45}{32}$$

$$x_2=\frac{3}{10}(\frac{5}{3}+\frac{2}{3}\times\frac{45}{32})=\frac{25}{32}$$

$$x_1=\frac{1}{3}(1+4\times \frac{45}{32}-2\times \frac {25}{32})=\frac {27}{16}$$

Answer.

$$x_1=\frac {27}{16}$$

$$x_2=\frac{25}{32}$$ $$x_3=\frac{45}{32}$$