Given the curve y=x^2 +1 and the line y=k, find, using graphical methods only, the value of k for which there is exactly one solution.
Graphically, the solution of a pair of simultaneous equations is the point of intersection between both curves. If we want only one solution to the system of equations, then there should only be one point of intersection between both lines. By drawing the graphs, we can see that y=k is a horizontal line, and y=x2+1 is a parabola with vertex at (0,1). Therefore, the only value of k for which a horizontal line crosses the parabola only once is y=k=1.
