Question-6

Suppose a system of linear equations consists of only one equation and four variables:

\[ x_{1}+x_{2}+x_{3}+x_{4}=a \]

where \(a\) is a constant. Find out the number of independent variables and find all possible solutions to this system.

---

We can form the matrix corresponding to this:

\[ \begin{bmatrix}\boldsymbol{1} & 1 & 1 & 1\end{bmatrix} \]

The first column is a pivot column. Hence, \(x_{1}\) is a dependent variable. \(x_{2},x_{3},x_{4}\) are independent variables. So this equation has three independent variables. To solve the system, we give arbitrary values to the independent variables and then solve for the dependent variable:

\[ \begin{aligned} x_{2} & =t_{2}\\ x_{3} & =t_{3}\\ x_{4} & =t_{4}\\ \implies x_{1} & =a-(t_{2}+t_{3}+t_{4}) \end{aligned} \]

Therefore, the set of all solutions to this system can be represented by this set:

\[ S=\left\{ \left(a-(t_{2}+t_{3}+t_{4}),t_{2},t_{3},t_{4}\right)\,:\,t_{2},t_{3},t_{4}\in\mathbb{R}\right\} \]