Video: Solving Systems of Linear and Quadratic Equations

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

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Video Transcript

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

What we have here is a linear-quadratic system of equations. The first equation π‘₯ plus 𝑦 equals six is linear. And the second equation π‘₯ squared plus π‘₯𝑦 equals 18 is quadratic because it includes an π‘₯ squared term and also the term π‘₯𝑦, 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 π‘₯. So we’re going to do this using the substitution method.

We begin by rearranging the first equation to give 𝑦 equals six minus π‘₯ because this gives an expression for 𝑦 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 𝑦 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 aren’t asked to find the value of 𝑦 in this problem. But if we did need to, we could substitute the value of π‘₯ we’ve just found back into our linear equation, 𝑦 equals six minus π‘₯, to find the corresponding value of 𝑦. Our solution to the problem is that the value of π‘₯ is three.

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