Video Transcript
In this video, we will learn how to
solve linear-quadratic systems of equations. Let’s begin by discussing what a
linear-quadratic system of equations actually is. It’s not as scary as it sounds. It’s a system of two equations in
which one is a linear equation and the other is a quadratic.
Remember, a linear equation is one
in which the highest power of each variable that appears is one and none of the
variables are multiplied together. For example, the equation 𝑦 equals
two 𝑥 is a linear equation. And if you were to draw its graph,
it would be represented by a straight line. A quadratic equation, however, is
one in which there will be a least one squared term. For example, the equation 𝑥
squared plus 𝑦 squared equals five is a quadratic equation. We may also see equations which
include terms when the two variables are multiplied together, for example, the
equation 𝑥 plus two 𝑥𝑦 equals three.
If you were to plot the graph of a
quadratic equation, it gives some kind of curve. In the case of 𝑥 squared plus 𝑦
squared equals five, it is a circle. From a graphical point of view,
solving a linear-quadratic system of equations is equivalent to finding the
coordinates of any points of intersection between the two graphs.
Now, we’re going to mostly be using
the method of solving simultaneous equations by substitution. So you should make sure that you’re
already familiar and comfortable with this method. You should also be comfortable
solving quadratic equations in one variable by factoring. We’ll look at some applications of
these techniques to worded problems and problems involving points of intersection of
straight lines and curves. Let’s consider our first example
then.
Solve the simultaneous
equations 𝑦 equals 𝑥 minus two, 𝑥 minus two squared plus 𝑦 minus three
squared is equal to nine.
The first thing to notice is
that this is a linear-quadratic system of equations. The first equation 𝑦 equals 𝑥
minus two is a linear equation, as the highest power of 𝑥 and 𝑦 that appears
is one. And the second equation 𝑥
minus two squared plus 𝑦 minus three squared equals nine is a quadratic
equation because once we’ve distributed the parentheses, we will have both 𝑥
squared and 𝑦 squared terms.
We’re going to use the method
of substitution to answer this problem. Now, our first equation is 𝑦
equals 𝑥 minus two. And we notice that the
expression 𝑥 minus two appears in the second equation. So what we can do is replace 𝑥
minus two with 𝑦 in our second equation. Doing so gives 𝑦 squared plus
𝑦 minus three squared equals nine. And so we have an equation in
𝑦 only. It’s a quadratic equation which
we can solve.
We begin by distributing the
parentheses in 𝑦 minus three squared. And we remember that 𝑦 minus
three all squared means 𝑦 minus three multiplied by 𝑦 minus three. So when we distribute the
parentheses, we have four terms. And these four terms then
simplify to 𝑦 squared minus six 𝑦 plus nine. Collecting like terms on the
left-hand side, we have the quadratic equation two 𝑦 squared minus six 𝑦 plus
nine equals nine.
Now, we notice that as we have
positive nine on each side, these two terms will directly cancel one another
out, which is equivalent to subtracting nine from each side of the equation. This leaves us with the
simplified equation two 𝑦 squared minus six 𝑦 equals zero, which we can solve
by factoring. The highest common factor of
two 𝑦 squared and negative six 𝑦 is two 𝑦. And to make two 𝑦 squared, we
have to multiply two 𝑦 by 𝑦. And to make negative six 𝑦, we
have to multiply two 𝑦 by negative three. So our quadratic in its
factored form is two 𝑦 multiplied by 𝑦 minus three is equal to zero.
To solve, we take each factor
in turn, set it equal to zero, and then solve the resulting linear equation. The first equation is two 𝑦
equals zero, which we can solve by dividing each side by two to find that 𝑦 is
equal to zero. The second equation is 𝑦 minus
three equals zero. We solve by adding three to
each side giving 𝑦 equals three. So we find that our solution to
these simultaneous equations includes two 𝑦-values, 𝑦 equals zero and 𝑦
equals three.
We now need to find the
corresponding 𝑥-values. And to do this, we substitute
each 𝑦-value into the linear equation. It must be the linear equation
into which we substitute because it’s possible that, on the quadratic curve,
there is more than one point where 𝑦 equals zero or 𝑦 equals three. But on the linear graph, the
straight line, there will only be one point where 𝑦 equals zero and one point
where 𝑦 equals three. So we must substitute into the
linear equation.
When 𝑦 equals zero, we have
zero equals 𝑥 minus two. And adding two to each side, we
find that 𝑥 is equal to two. When 𝑦 equals three, we have
that three is equal to 𝑥 minus two. And adding two to each side, we
find that 𝑥 is equal to five. So the solution to this pair of
simultaneous equations is two pairs of 𝑥𝑦-values. 𝑥 equals two and 𝑦 equals
zero and 𝑥 equals five and 𝑦 equals three.
It’s important to understand
that these solutions come in pairs. And we can’t mix and match the
𝑥- and 𝑦-values. For example, 𝑥 equals two and
𝑦 equals three is not a valid solution to this pair of simultaneous equations,
which you can confirm by substituting the values into the linear equation or,
indeed, the quadratic.
Remember, we used the method of
substitution to answer this question by replacing 𝑥 minus two with 𝑦. It would also have been
possible to replace 𝑦 in the second set of parentheses with 𝑥 minus two to
give an equation in 𝑥 only. In this case, we’d have found
our solutions for 𝑥 first and then substitute it back into the linear equation
to find our solutions for 𝑦. Our final answer is that 𝑥
equals two and 𝑦 equals zero or 𝑥 equals five and 𝑦 equals three.
In our next example, we’ll see how
we can apply these methods to a worded problem.
The sum of two numbers is 11
and the sum of their squares is 65. What are the numbers?
Now, it’s important to notice
that we shouldn’t answer this question using trial and error. We need to use a formal
approach. We’re going to use some algebra
to answer the problem. So we’ll let the two numbers be
represented by the letters 𝑥 and 𝑦. We’re now going to express the
information in the question as equations involving 𝑥 and 𝑦. Firstly, the sum of the two
numbers is 11. So this gives us the equation
𝑥 plus 𝑦 is equal to 11.
Secondly, we’re told that the
sum of their squares is 65. So this gives us the equation
𝑥 squared plus 𝑦 squared equals 65. We now have a linear-quadratic
system of equations. The first equation is linear,
and the second is quadratic. We’re going to use the method
of substitution to solve these two equations simultaneously. We begin by rearranging the
linear equation to give one variable in terms of the other. And in this problem, it doesn’t
make any difference which variable we choose because the problem is equally
complicated or equally simple in both variables. I’ve chosen to rewrite equation
one as 𝑦 equals 11 minus 𝑥.
We now take this expression for
𝑦 in terms of 𝑥 and substitute it into the second equation, that’s into the
quadratic equation. Giving 𝑥 squared plus 11 minus
𝑥 all squared is equal to 65. And now, we have a quadratic
equation in one variable 𝑥 only. We can distribute the
parentheses carefully, remembering that 11 minus 𝑥 all squared means 11 minus
𝑥 multiplied by 11 minus 𝑥, which simplifies to 121 minus 22𝑥 plus 𝑥
squared. We can group the like terms on
the left-hand side — that’s 𝑥 squared plus 𝑥 squared — giving two 𝑥
squared. And at the same time, subtract
65 from each side to give the quadratic equation two 𝑥 squared minus 22𝑥 plus
56 is equal to zero.
Now, we notice at this point
that each of the coefficients in our equation are even numbers. So we can simplify by dividing
the entire equation by two. Doing so gives the simplified
quadratic equation 𝑥 squared minus 11𝑥 plus 28 is equal to zero. Now, we want to solve this
equation for 𝑥. So we first look to see whether
this equation can be factored. As the coefficient of 𝑥
squared in our equation is one, the first term in each set of parentheses will
simply be 𝑥 because 𝑥 multiplied by 𝑥 gives 𝑥 squared. And to complete the
parentheses, we’re then looking for two numbers whose sum is the coefficient of
𝑥 — that’s negative 11 — and whose product is the constant term — that’s
28.
By considering the factors of
28, we see that if we choose the two numbers negative seven and negative four,
then their product is indeed 28 because a negative multiplied by a negative
gives a positive. And their sum is indeed
negative 11. So these are the two numbers
we’re looking for to complete our parentheses. We have our quadratic in its
factored form then. 𝑥 minus seven multiplied by 𝑥
minus four is equal to zero.
To solve, we take each factor
in turn, set it equal to zero, and solve the resulting equation. We have 𝑥 minus seven equals
zero, which can be solved by adding seven to each side to give 𝑥 equals seven,
and then 𝑥 minus four equals zero, which we solve by adding four to each side
to give 𝑥 equals four. We found then the solution for
𝑥. There are two possible
values. 𝑥 is either equal to four or
𝑥 is equal to seven.
We now need to find the
corresponding 𝑦-values, which we can do by substituting each 𝑥-value in turn
into the linear equation which, remember, that was 𝑦 is equal to 11 minus
𝑥. When 𝑥 is equal to seven, we
find that 𝑦 is equal to 11 minus seven which is four. And when 𝑥 is equal to four,
we find that 𝑦 is equal to 11 minus four, which is seven. So we have exactly the same
values for 𝑦, as we did for 𝑥.
The reason for this is because
at the beginning of the problem, we just let the two numbers be represented by
𝑥 and 𝑦. We didn’t specify whether 𝑥 or
𝑦 was the larger number. So we found the same solution
twice. The two numbers are seven and
four. And either can be 𝑥 and then
the other will be 𝑦. We can of course check our
answer, firstly, by confirming that the sum of our two numbers is 11, which of
course it is, and, secondly, by confirming that the sum of their squares — seven
squared plus four squared, which is 49 plus 16 — is indeed equal to 65.
So by first formulating the
information in the question as a linear-quadratic system of equations and then
solving this pair of simultaneous equations using the substitution method, we
found that the two numbers we’re looking for are seven and four.
Now, one really useful application
of these techniques is in finding the points of intersection of a straight line and
a curve, as mentioned right at the start of the video. Let’s now consider an example of
this type.
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 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.
In our final example, we’ll
consider a problem where the two variables are multiplied together in one of the two
equations.
Given that 𝑥 squared plus 𝑥𝑦
equals 18 and 𝑥 plus 𝑦 equals six, find the value of 𝑥.
What we have here is a
linear-quadratic system of equations. The first equation 𝑥 plus 𝑦
equals six is linear. And the second equation 𝑥
squared plus 𝑥𝑦 equals 18 is quadratic because it includes an 𝑥 squared term
and also the term 𝑥𝑦, where the two variables are multiplied together. We’re not asked to fully solve
this system of equations but simply to find the value of 𝑥. So we’re going to do this using
the substitution method.
We begin by rearranging the
first equation to give 𝑦 equals six minus 𝑥 because this gives an expression
for 𝑦 in terms of 𝑥, which we can substitute into the second equation to give
an equation in 𝑥 only. Doing so gives the equation 𝑥
squared plus 𝑥 multiplied by six minus 𝑥 equals 18. And we now have a quadratic
equation in 𝑥, which we can solve.
We distribute the parentheses
on the left-hand side to give 𝑥 squared plus six 𝑥 minus 𝑥 squared equals
18. And we now see that the 𝑥
squared and negative 𝑥 squared terms will cancel each other out. So, in fact, our equation
reduces to a linear equation. We have six 𝑥 is equal to
18. And this equation can be solved
by dividing each side by six to give 𝑥 equals three. So by substituting 𝑦 equals
six minus 𝑥 into the second equation, we created an equation in 𝑥 only which
we could then solve to find the value of 𝑥.
Of course, we aren’t asked to
find the value of 𝑦 in this problem. But if we did need to, we could
substitute the value of 𝑥 we’ve just found back into our linear equation, 𝑦
equals six minus 𝑥, to find the corresponding value of 𝑦. Our solution to the problem is
that the value of 𝑥 is three.
Let’s now summarize what we’ve seen
in this video. Firstly, a linear-quadratic system
of equations is simply a system of two equations in which one is linear and the
other is quadratic. We solve such systems of equations
by using the method of substitution. We rearrange the linear equation to
give one variable in terms of the other, and then we substitute our expression for
this variable into the quadratic equation. We then solve the resulting
quadratic equation, usually by factoring. We then substitute the value or
values we found for the first variable back into the linear equation to find the
corresponding values for the second.
We must remember that our solutions
come in pairs. So we must give our solution as
corresponding pairs of the two variables. We can’t mix and match the
different values. We saw also in this video that this
method can be applied to number problems. And a particularly useful
application is in finding the coordinates of any points of intersection between a
straight line and a curve.
