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Section 1.2 Equations with multiple solutions or no solutions

Subsection 1.2.1 Equations with no solutions

Suppose we want to find all solutions to the equations

\begin{align*} 2x+4y \amp = 5\\ -x-2y \amp = 1 \end{align*}

Using the standard approach we multiply the second equation by \(2\) and add it to the first one to eliminate the variable \(x\text{.}\) This leaves us with the equation 0=7 This is certainly an equality that is not valid. What happened? We can see by multiplying the second equation by \(-2\text{.}\) We then have

\begin{align*} 2x+4y \amp = 5\\ 2x+4y \amp = -2 \end{align*}

Whatever \(x\) and \(y\) are, the value of \(2x+4y\) can't be \(5\) and \(-2\) at the same time. So there are no solutions. What happens if we try to graph these two equations? Here is what we get:

Figure 1.2.1. Two Parallel lines in the Plane

The geometry of the situation is now clear: the two lines are parallel and so there is no point on both lines (indeed, both lines have slope \(-\tfrac12\)). When we have equations with no common solution, they are called inconsistent.

Subsection 1.2.2 Equations with more than one solution

Now let's alter the equations of Subsection 1.1.1 slightly. We consider the pair of equations

\begin{align*} 2x+4y \amp = -2\\ -x-2y \amp = 1 \end{align*}

We apply the standard method again: multiply the second equation by \(2\) and add it to the first. The result is

\begin{equation*} 0=0 \end{equation*}

This is certainly a valid, although not very interesting, equation. In fact, if we multiply both sides of the second equation by \(-2\) the system becomes

\begin{align*} 2x+4y \amp = -2\\ 2x+4y \amp = -2 \end{align*}

This means that any solution of the first equation is also a solution of the second one. Geometrically, if we plot the graph of the two equations, the same line results for each one. How do we find all solutions in this case? Let us assign a value to \(y\text{.}\) Let's call it \(t\) so \(y=t\text{.}\) Then, using either equation, we have \(x=-2t-1\text{.}\) This means that for any value of \(t\) we know that \((x,y)=(-2t-1,t)\) is a solution to both equations. So, in fact we have an infinite number of solutions.

The corresponding lines in the plane are parallel, intersect at a single point, or are identical.