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 𝑦.
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
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.