Solution

Elementary Linear algebra

Chapter 1.2

Exercise 4

$t$, where $t$ is a real parameter, the solution is given by parametric equations:

$$x_1 = 8+7t, x_2 = 2-3t, x_3 = -5-t, x_4 = t$$

Introduce $x_2 = t$ and $x_5 =s$. The solution set is given by parametric equations:

$$x_1 = 6t-3s-2, x_2=t,x_3=-4s+7, x_4 = -5s+8, x_5=s$$

(d) Note that the given system is inconsistent. The last row of the matrix indicates

$$0x_1+0x_2+0x_3 = 1$$

(a)

$$x_1=-3, x_2=0,x_3=7$$

(b)

$$x_1 = 8+7t, x_2 = 2-3t, x_3 = -5-t, x_4 = t$$

(c)

$$x_1 = 6t-3s-2, x_2=t,x_3=-4s+7, x_4 = -5s+8, x_5=s$$

(d) The system has no solutions, i.e. it is inconsistent.