Solving linear equations and inequalities: An equation of a straight line in a plane
Solution of a linear equation with two unknowns
In the discussion about linear equations we have seen that the equation \(ax+b =0\) has solution \(x=-\frac{b}{a}\), at least, if \(a\ne0\). Herein \(a\) and \(b\) represent real numbers, for which we often have substituted concrete values. But as we have already done before, we can even write this equation and the solution with unknowns \(a\) and \(b\) in this general form. When we do that, then \(a\) and \(b\) are called parameters. In general, parameters are variables that occur in mathematical expressions, but do not play a role as unknowns in an equation. They represent distinghuised numbers that have not yet been specified.
It is however important that we mark \(x\) as unknown. We could also have considered \(a\) as unknown, in which case the solution would have been \(a=-\frac{b}{x}\), at least, when \(x\ne0\) .
In this way we can also solve an equation with two unknowns.
The linear equation \(ax+by=c\), with unknowns \(x\) and \(y\), in which \(a\), \(b\) and \(c\) are numbers or parameters, can be solved in the following two ways:
- By considering \(y\) temporarily as parameter: when we solve the linear equation with unknown \(x\), we get \(x=-\frac{b}{a}y+\frac{c}{a}\) . This is only possible if \(a\ne0\); the solutions are all dyads \([x,y]\) of the form \([-\frac{b}{a}y+\frac{c}{a},y]\). Here, the role of \(y\) as a parameter is clear: for each value of \(y\), there is exactly one solution.
- considering \(x\) temporarily as parameter: when we solve the linear equation with unknown \(y\), we get \(y=-\frac{a}{b}x+\frac{c}{b}\). This is only possible if \(b\ne0\); the solutions are all dyads \([x,y]\) of the form \([x,-\frac{a}{b}x+\frac{c}{b}]\). Here, the role of \(y\) as a parameter is clear: for each value of \(x\), there is exactly one solution.
If \(a\ne0\) and \(b\ne0\), then there is an overlap between the first and second case. Each of the two gives a way to describe the set of solutions. The first does so by considering \(x\) as a function of \(y\), the second by considering \(y\) as a function of \(x\). The vertical line at \(a=0\) occurs exclusively in the first case, and the horizontal line at \(b=0\) occurs only in the second case.
We now have identified all possible solutions when \(a\ne0\) or \(b\ne0\). The results form the set of points on a line in the plane.
If \(a=0\) and \(b=0\), then the equation is \(0=c\). So two cases are left:
- if \(c=0\), then each dyad \([x,y]\) is a solution, and
- if \(c\ne0\), then there is no solution.
We proceed as in solving a linear equation with unknown \(y\). Thus, we consider \(x\) as a parameter.
\[ \begin{array}{rclcl} 5 x+7 y&=&10&\phantom{xxxxx}&\blue{\text{the given equation}}\\
7\cdot y &=& 10- 5 x&\phantom{xxxxx}&\blue{\text{movement of terms without }y\text{ to the right}}\\
y &=& \dfrac{10-5x}{7} &\phantom{xxxxx}&\blue{\text{division by the coefficient }7\text{ of }y} \\
y &=& -{{5}\over{7}} x + {{10}\over{7}}&\phantom{xxxxx}&\blue{\text{simplified order of terms}}
\end {array}\]
