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Find the set of points of intersection of the graphs of 𝑦 equals three 𝑥 and 𝑥 squared plus 𝑦 squared equals 40.
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Well, the first thing to look at is points of intersection.
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So what does this actually mean?
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But I think a little sketch is gonna help us understand.
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So what am I drawing is a rough sketch of our two graphs.
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So we got our 𝑦 equals three 𝑥 and our 𝑥 squared plus 𝑦 squared equals 40.
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Well, the points of intersection are in fact the places where these two graphs would meet.
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So with these graphs, we can see that there will be in fact two points of intersection.
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And these are where those two points are.
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So now we know that we’re gonna be looking for two points of intersection.
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And we know what points of intersection are.
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Let’s go about solving this problem.
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So to do that, we’re gonna write down our two equations.
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And as you can see, I’ve actually labelled them.
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So we’ve got equation one and equation two, or graph one and graph two.
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So what we said before is there are points of intersection where the two graphs meet.
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So what we want to do is actually make our two equations equal to each other.
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And the way we can do that is by using the substitution method.
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So to enable us to do that, what I’m gonna do is I’m gonna substitute our value for 𝑦 so that 𝑦 is equal to three 𝑥 into equation two.
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So we’re gonna get 𝑥 squared plus three 𝑥 all squared.
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And this is three 𝑥 all squared because we’ve substituted our three 𝑥 for our 𝑦.
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And this is all gonna be equal to 40.
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So then we’re gonna get 𝑥 squared plus nine 𝑥 squared is equal to 40.
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Being careful of the three 𝑥 all squared is three 𝑥 multiplied by three 𝑥, so you get nine 𝑥 squared not just three 𝑥 squared.
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So then we can collect our like terms.
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We can collect our 𝑥 squared terms.
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So we’re gonna get 10𝑥 squared is equal to 40.
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And then we divide both sides by 10 which gives us that 𝑥 squared is equal to four.
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And then we’re gonna take the square root of both sides.
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So we’re gonna get that 𝑥 is equal to negative two or two.
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And the reason it’s negative two or two is cause if we look back at our graph in the top right-hand side, we know we’re gonna have two solutions.
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So we’re gonna have two different values of 𝑥.
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So there now are two values of 𝑥.
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And now we’re gonna find our 𝑦-coordinates or 𝑦-values.
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And the way we’re gonna do that is by substituting our 𝑥-values back into equation one.
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You could actually do it into either equation.
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I’ve just chosen to do it into equation one.
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Looks that’ll be the simplest in this question.
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So when 𝑥 is equal two, we’re gonna get 𝑦 is equal to three multiplied by two cause we’ve substituted in two for our 𝑥-value.
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So we can get a 𝑦-value, a 𝑦-coordinate of six.
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Or if 𝑥 is equal to negative two, then we’re gonna substitute in negative two for our 𝑥-value.
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So we’re gonna get 𝑦 is equal to three multiplied by negative two.
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So we’re gonna get our 𝑦-value or 𝑦-coordinate of negative six.
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So therefore, we can say that our set of points of intersection are: two, six and negative two, negative six.
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And I’ve shown that using our set notation to identify that.
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So just a quick recap of what we’ve done.
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So first of all, we used the substitution method.
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So we substituted 𝑦 equals three 𝑥 into equation two.
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Then we’ve solved that to find our 𝑥-values which there were two of, cause we knew there’d be two values because of our little sketch.
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And then we substitute those 𝑥-values back into one of our equations in order to find our 𝑦-values, 𝑦-coordinates.