Solve each problem using a system of equations in two variables. ...

Question

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose squares have a sum of 100 and a difference of 28.

Accepted Answer

Hello, everybody. I hope we're doing all right. Today, today we're going to be looking at this math question that states using a system of nonlinear equations into variables determine the two numbers whose sum of squares is equal to 1 30 whose difference of squares is equal to 32. Now, the Tetris provided are a plus or minus nine and plus or minus seven B plus or minus eight and plus or minus seven C plus or minus six N plus or minus nine D plus or minus eight and plus or minus six. Now we look at the information that they gave us, they gave us a sum of squares, which would be X squared plus Y squared. That is a sum of squares and that sum of squares is equal to 130. And in this case, I used X and Y uh but you could use any two variables. Now, they also told us that the difference of squares of those same two variables, variables. So X squared minus Y squared, that difference of squares is 32. No, I, I wrote the sum of squares on top of the difference of squares. Because if you write it like that, you can visualize it as adding the two fractions up. So that usually when you're adding two things, you're, we're always taught to, you write them on top of each other, you write the plus sign and then you write a line to add them up. So that's why I wrote them one on top of the other. So once, once we add up both of these equations, it's kind of like using elimination, right, we're eliminating one of the variables here because once we add them up X squared plus X squared will be two X squared. And then we have a positive Y square on top and a negative Y squared on the bottom. So we have P A Y squared minus Y squared. So we get rid of the Y variable and finally, we'll have 130 minus or plus the which is 100 and 62. So when, once we add up both of our equations to use the elimination method, we'll get two X squared equals 100 and 62. And now we can solve for X. So if we divide both sides by two, we'll have the X squared is equal to 81. And then to get rid of the X to the, of the square, the second power we take the square root of both sides. So we have that X is equal to the square root of or X is equal to plus or minus nine. So it's plus or minus because remember if you have any negative nine squared, it'll still be positive 81. So it could either be a positive nine or a negative nine as our answer for X. So one of our answers in this case, the X is plus or minus nine. Now I can plug that X into one of our equations. In my case, I'll be plugging it, plugging it into the sum of the squares. So X squared plus Y squared equals 130. And you could use either. So normally you have, you would have to plug in the positive nine and the negative nine to see what answers each of those would give. However, because our X is squared here, even if we use the negative nine, it'll still turn into a positive answer. So, so either or would work. I'm just going to use the positive nine for, for my problem, but you could use the negative nine and it'll give you the same answer. So I'll plug in the positive nine for my X. So we'll have nine squared plus Y squared equals 130 which is 81 plus Y squared equals 130. And then I can move the 81 over from the left hand side to the right hand side to have Y squared equals 100 and 30 minus 81 which is 49. And then again, to get rid of the second power, we take the square root of both sides. So I have that Y is equal to the square root of or Y is equal to plus or minus seven. Once again, the plus or minus B because if you square a negative seven, you still get a positive 49. So our answers will be plus or minus nine and plus or minus seven or answer choice A in this case. So I hope that was helpful. And until next time.

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose squares have a sum of 100 and a difference of 28.

Key Concepts

System of Equations

Quadratic Relationships

Difference and Sum of Squares

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