### Systems of linear equations: Basic concepts and methods

### Solving a linear equation with several unknowns

In the same way as we previously described the general solution of a linear equation with a single unknown, we can also solve a linear equation with two or more unknowns; it is only a matter of selecting unknowns for their roles as temporary parameters.

First, we discuss equations with two unknowns.

Solving the linear equation with two unknowns The linear equation \(a\, x+b\, y=c\), with unknown \(x\) and \(y\) where \(a\), \(b\), and \(c\) are numbers of parameters, are numbers or parameters, can be solved in the following two ways:

*Viewing \(y\) as a temporary parameter*we solve the linear equation with unknown \(x\), and conclude that \(x=-\frac{b}{a}\, y+\frac{c}{a}\). This is possible if only if \(a\ne0\); therefore the solutions are all pairs \([x,y]\) of the form \([-\frac{b}{a}\, y+\frac{c}{a},y]\). Here, the role of \(y\) as a parameter becomes clear: for each value of \(y\) there is exactly one solution.*Viewing \(x\) as a temporary parameter*we solve the linear equation with unknown \(y\), and conclude that \(y=-\frac{a}{b}\cdot x+\frac{c}{b}\). This is possible if and only if \(b\ne0\); therefore, the solutions are all pairs \([x,y]\) of the form \([x,-\frac{a}{b}\, x+\frac{c}{b}]\). Here, the role of \(x\) as a parameter becomes clear: for each value of \(x\) there is exactly one solution.

If both \(a\ne0\) and \(b\ne0\), then there is overlap between the first and second case. Each of the two provides a way to describe the set of solutions. The former does so by viewing \(x\) as a function of \(y\), the latter by viewing \(y\) as a function of \(x\). Exclusively for the first case, the vertical line for \(a=0\) occurs, and for the second case, the horizontal line for \(b=0\) occurs.

We proceed as in solving a linear equation with unknown \(y\). Thus, we consider \(x\) as a parameter.

\[ \begin{array}{rclcl} 6 x+6 y&=&4&\phantom{xxxxx}&\blue{\text{the given equation}}\\

6\cdot y &=& 4- 6 x&\phantom{xxxxx}&\blue{\text{terms without }y\text{ moved to the right}}\\

y &=& \displaystyle 6 x+6 y=4&\phantom{xxxxx}&\blue{\text{divided by the coefficient }6\text{ of }y} \\

y &=&\displaystyle -x + {{2}\over{3}}&\phantom{xxxxx}&\blue{\text{simplified order of terms changed}}

\end {array}\]

Remains the general case.

Isolating an unknown in a linear equation with \(n\) unknowns Consider the linear equation \[a_1x_1 + \cdots + a_nx_n + b = 0\] where \(a_1, \ldots, a_n\) and \(b\) are real or complex numbers and \(x_1, \ldots, x_n\) unknowns.

- Suppose that the coefficient of one of the unknowns is distinct from zero, say \(a_1\neq 0\); then the unknown \(x_1\) can be isolated by viewing all the other unknowns as temporary parameters: \[ x_1=\frac{-b - a_2x_2 -\cdots - a_nx_n}{a_1}\]
- If
*all*coefficients are zero, there remain two cases:

- if \(b=0\), then each list of \(n\) numbers \(\rv{x_1,\ldots,x_n}\) is a solution;
- if \(b\ne0\), then there is no solution