Example 5.1.18 Coinciding Lines

Next we'll look at the other special case. Let's start with this system of equations:

\begin{equation*} \left\{ \begin{alignedat}{4} y \amp {}={} \amp 2x-4 \\ 6x-3y \amp {}={} \amp 12 \end{alignedat} \right. \end{equation*}

To solve this system of equations, we want to graph each line. The first equation is in slope-intercept form and can be graphed easily using its slope of \(2\) and its \(y\)-intercept of \((0,-4)\text{.}\)

The second equation, \(6x-3y=12\text{,}\) can either be graphed by solving for \(y\) and using the slope-intercept form or by finding the intercepts. If we use the intercept method, we'll find that this line has an \(x\)-intercept of \((2,0)\) and a \(y\)-intercept of \((0,-4)\text{.}\) When we graph both lines we get FigureĀ 5.1.19.

Now we can see these are actually the same line, or coinciding lines. To determine the solution to this system, we'll note that they overlap everywhere. This means that we have an infinite number of solutions: all points that fall on the line. It may be enough to report that there are infinitely many solutions. In order to be more specific, all we can do is say that any ordered pair \((x,y)\) satisfying the line equation is a solution. In set-builder notation, we would write \(\{(x,y)\mid y=2x-4\}\text{.}\)

Figure 5.1.19 Graphs of \(y=2x-4\) and \(6x-3y=12\)