Example 5.1.16 Parallel Lines

Let's look at the graphs of two lines with the same slope, \(y=2x-4\) and \(y=2x+1\text{:}\)

a coordinate grid with two parallel lines; one line has a y-intercept of -4 and the other has a y-intercept of 1; they both have a slope of 2
Figure 5.1.17 Graphs of \(y=2x-4\) and \(y=2x+1\)

For this system of equations, what is the solution? Since the two lines have the same slope they are parallel lines and will never intersect. This means that there is no solution to this system of equations. We write the solution set as \(\emptyset\text{.}\)