Simultaneous Equations Question(Sorry if it’s really simple)

[Guest]

I have a quick Simultaneous Equations question, and it might be very simple, but I just need help getting from one step to the next here.

The question is "If x and y are positive numbers such that x+y=1, which of these could be the value os 100x+200y?"

Then they give a bunch of values. I understand how to test each value once they give me the final combined equation, but I just don't know how they got from one to the other.

It says "Since x+y=1, then y=1-x" that I get. Then they say "y=1-x and 100x+200y can be expressed as 100x+200(1-x)=200-100x"

I need that jump explained please. I feel like it's going to be super simple once I see it, I just need it spelled out. My brain isn't working right now, haha.

THANK YOU TO ANYONE WHO ANSWERS THIS!

[Guest]

Could you post the whole(original) question, please?

[HallsofIvy]

You say you "get' that x+ y= 1 leads to y= 1- x (subtract x from both sides of the first equation). That says that "y" and "1- x" are different ways of writing the same number so we can replace the "y" in 100x+ 200y with "1- x". That is, 100x+ 200y= 100x+ 200(1- x). Now use what is technically called the "distributive law": for any number, a, b, and c, a(b+ c)= ab+ ac. 200(1- x)= 200- 200x. So 100x+ 200y= 100x+ 200(1- x)= 100x+ 200(1)- 200x= 200- 100x (because, of course, 100x- 200x= -100x).

Notice that we could have just as easily have said that, since x+ y= 1, x= 1- y. Then 100x+ 200y= 100(1- y)+ 200y= 100- 100y+ 200y= 100+ 100y, writing this in terms of y rather than x.

An equivalent way to do that would be to write 100x+ 200y= 100x+ 100y+ 100y= 100(x+ y)+ 100y and, since x+ y= 1, 100x+ 200y= 100+ 100y.