Find the set of points of
intersection of the graphs of 𝑥 minus 𝑦 equals zero and six 𝑥 squared minus
𝑦 squared equals 45.
So we’ve been asked to find the
points of intersection of 𝑥 minus 𝑦 equals zero, which is a straight line, and
six 𝑥 squared minus 𝑦 squared equals 45, which is a curve. This is equivalent to solving
the linear-quadratic system of equations 𝑥 minus 𝑦 equals zero, six 𝑥 squared
minus 𝑦 squared equals 45. We’re going to do this using
the method of substitution. We begin by rearranging the
linear equation to give one variable in terms of the other. We find that 𝑥 is equal to
We’re now going to substitute
our expression for 𝑥 into the second equation. Doing so gives six 𝑦 squared
minus 𝑦 squared is equal to 45. We could equally have
substituted 𝑦 equals 𝑥 into our second equation, which would give six 𝑥
squared minus 𝑥 squared equals 45. Both approaches will lead us to
the same solution. We can now solve this quadratic
equation for 𝑦.
Simplifying the left-hand side,
six 𝑦 squared minus 𝑦 squared, gives five 𝑦 squared. We can then divide through by
five, giving 𝑦 squared equals nine and solve by square rooting. Remembering, we must take plus
or minus the square root. So we have that 𝑦 is equal to
plus or minus the square root of nine, which is positive or negative three.
Having found our 𝑦-values, we
now need to find the corresponding 𝑥-values by substituting into the linear of
equation. And it’s very
straightforward. As our linear equation can be
expressed as 𝑥 equals 𝑦, then each 𝑥-value is just the same as the
corresponding 𝑦-value. We find then that there are two
points of intersection between these graphs, the point three, three and the
point negative three, negative three, which we can express as the set containing
these two coordinates.
Now, you may not immediately
recognize what the graph of six 𝑥 squared minus 𝑦 squared equals 45 looks
like. But if you have access to a
graphics calculator or some graphical plotting software, then you can plot these
two graphs. 𝑥 minus 𝑦 equals zero is a
straight line and six 𝑥 squared minus 𝑦 squared equals 45 is what’s known as a
hyperbola. And by considering these two
graphs, you can confirm that the two points of intersection are indeed the
points we’ve given here.