In Exercises 1–18, solve each system by the substitution method. ...

Question

In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x^2−4

Accepted Answer

Welcome back. I am so glad you're here. We are given these two equations. The first one here is x minus two, Y equals four. And the second one is X equals y squared minus four. We asked to solve the system of equations using substitution. Alright, We've already got an X equal to something for the second equation. So we can put that something the y squared minus four in for X in the first equation. So now that will read y squared minus four minus two, Y equals four for that first equation. And now we can solve for why? Well we can do a few things here, we can bring this four over from the right hand side to the left hand side so that it will be equal to zero and we'll have a quadratic and hopefully we can factor it. And also I'm going to rewrite these in order of their descending degrees. So starting with Y squared and then minus two Y. And then we have negative 4 -4. So that's -8 equals zero on the right hand side. So now we can factor this what times what gives us why? Square? That's why times why? What? Two numbers when multiplied, give us a negative eight. But when added together give us a negative two, that's going to be a minus four plus two. And now we have these two factors and we can set them both equal 20 to solve for why? So why minus four equals zero will add four to both sides. And that will give us why equals four for one of our solutions. And for y plus two equals zero. We can subtract two from both sides and then we'll have Y equals negative two for our other Y solution. Now when we create these solution sets with our fancy dancy squiggly lines here, we know that our ordered pairs have an X. And then a Y value. And we've solved for our Y values. One of our y values is four and one is negative two. And now we just have to figure out what the corresponding X values are when Y equals four and negative two. So let's use equation too where X equals y squared minus four and let's first figure out when y equals four, what does X equal? So X equals four squared minus four. Will four squared is 16. So 16 minus four, X equals 12. When y equals four there's one of our X solutions. Now let's figure out what X equals when Y equals negative two. Using that same equation will take negative two squared minus four to solve for X and negative two squared that is four. So we have four minus four while X equals 01, Y equals negative two. So we look at our answer choices and although they've ordered their ordered pairs in a different order, this matches with answer choice. A Well done we'll catch you on the next one

In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x^2−4

Key Concepts

Systems of Equations

Substitution Method

Quadratic Functions

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