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
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.