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.