# Question Video: Solving Systems of Linear and Quadratic Equations Mathematics

Given that π₯Β² + π₯π¦ = 18, and π₯ + π¦ = 6, find the value of π₯.

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### Video Transcript

Given that π₯ squared plus π₯π¦ equals 18 and π₯ plus π¦ equals six, find the value of π₯.

What we have here is a linear-quadratic system of equations. The first equation π₯ plus π¦ equals six is linear. And the second equation π₯ squared plus π₯π¦ equals 18 is quadratic because it includes an π₯ squared term and also the term π₯π¦, where the two variables are multiplied together. Weβre not asked to fully solve this system of equations but simply to find the value of π₯. So weβre going to do this using the substitution method.

We begin by rearranging the first equation to give π¦ equals six minus π₯ because this gives an expression for π¦ in terms of π₯, which we can substitute into the second equation to give an equation in π₯ only. Doing so gives the equation π₯ squared plus π₯ multiplied by six minus π₯ equals 18. And we now have a quadratic equation in π₯, which we can solve.

We distribute the parentheses on the left-hand side to give π₯ squared plus six π₯ minus π₯ squared equals 18. And we now see that the π₯ squared and negative π₯ squared terms will cancel each other out. So, in fact, our equation reduces to a linear equation. We have six π₯ is equal to 18. And this equation can be solved by dividing each side by six to give π₯ equals three. So by substituting π¦ equals six minus π₯ into the second equation, we created an equation in π₯ only which we could then solve to find the value of π₯.

Of course, we arenβt asked to find the value of π¦ in this problem. But if we did need to, we could substitute the value of π₯ weβve just found back into our linear equation, π¦ equals six minus π₯, to find the corresponding value of π¦. Our solution to the problem is that the value of π₯ is three.