Question Video: Finding Unknown Coordinates of Points Using the Midpoint Formula Mathematics

Point (2, −7) is the midpoint of the line segment on endpoints (𝑥, −9) and (1, 𝑦). What is 𝑥 + 𝑦?

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Video Transcript

Point two, negative seven is the midpoint of the line segment on endpoints 𝑥, negative nine and one, 𝑦. What is 𝑥 plus 𝑦?

Well, with this question, what we’re actually dealing with is the midpoint of a line segment. Now the midpoint of a line is actually found by this following formula. Well, if we want to find the 𝑥-coordinate, it’s 𝑥 one plus 𝑥 two divided by two, and the 𝑦-coordinate is 𝑦 one plus 𝑦 two divided by two. And that’s when these are the actual coordinates of the points either end of our line segment.

Well, in our question, we’re actually told the endpoints. We’re told that they are 𝑥, negative nine and one, 𝑦. Therefore, I can label them 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two.

Now the point of this question is to actually find out what the value of 𝑥 plus 𝑦 is. But to enable us to do that, we first need to calculate 𝑥 and 𝑦. And the way we do that is by using our midpoint formula. So what I’m gonna do is actually start with the 𝑥-coordinate of our midpoint formula. And so if we got the 𝑥-coordinate, we can actually find that by 𝑥 one plus 𝑥 two over two. So therefore, what we can actually do is form an equation, because we know that 𝑥 plus one — and that’s because our 𝑥 one is 𝑥 and our 𝑥 two is one — divided by two is gonna be equal to two. And that’s because we’re told that the midpoint of the line has the 𝑥-coordinate two.

So therefore, if we actually multiply both sides of the equation by two to get rid of the fraction, what we’ll get is 𝑥 plus one is equal to four. That’s cause 𝑥 plus one over two multiplied by two just becomes 𝑥 plus one. And then two multiplied by two gives us four. And then if we actually subtract one from each side of the equation, what we’ll get is 𝑥 is equal to three. So therefore, we found 𝑥.

So great! Now let’s move on to our 𝑦-coordinate. Well, with our 𝑦-coordinate, what we’re gonna get is that negative nine, because this is our 𝑦 one, plus 𝑦, because this is our 𝑦 two, all divided by two is gonna be equal to negative seven. And that’s because that’s the 𝑦-coordinate of our midpoint of the line segment.

So again, what we’re gonna do is multiply each side of our equation by two. And we’re gonna do that as before to get rid of the fraction. So when we multiply both sides of the equation by two, we get negative nine plus 𝑦. And the left-hand side is equal to negative 14. That’s because negative seven multiplied by two is negative 14.

And then what we do is we add nine to each side of the equation. And so if we add nine to each side of the equation, well, on the left-hand side, we have negative nine add nine is zero. So we’re just left with 𝑦. And then on the right-hand side of the equation, we get negative 14. Add nine to negative 14, we get to negative five. So we got 𝑦 equals negative five.

So now what we’ve done is we’ve actually found 𝑥 equal to three and 𝑦 is equal to negative five. So have we finished? Is that the end of the question? Well, no, because the question actually asks us what is the value of 𝑥 plus 𝑦.

Well, if we wanna find the value of 𝑥 plus 𝑦, what we need to do is actually substitute in our values of 𝑥 and 𝑦. So what we’re gonna have is three plus negative five. Well, if we’ve got three plus negative five, this is the same as three minus five, which is gonna equal negative two. That’s because if you take five away from three, you get negative two. So therefore, we can say if the point two and negative seven is the midpoint of a line segment on endpoints 𝑥, negative nine and one, 𝑦, then 𝑥 plus 𝑦 is equal to negative two.

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