Question

I need help im a teenage student and I don't get this problem no one is helping me and I would appreciate if you helped me...

prove that the slope between any two points on the line 5x+y=7 is -5

Answer 1

Have you made any attempts at the problem so far?

Answer 2

yes but I keep getting it wrong

Answer 3

There are significant connection issues at the moment.
I don't know when you are returning, and I will be taking off. So I will post my best guide for my own thoughts.

Our goal is to prove for ANY two points on the line (5x + y = 7) the slope is equal to -5.
So, the word "ANY" is very important here. We can't just select any random two points and say, "we have sufficiently shown all points meet this requirement" when we only showed it was true for two. And we can't spend our entire life doing every single point on that line (there are an infinite amount). We'll have to get general. We need some defined variables.

Let's say we have two x-values.
x = a,
and x = b.
and we say that a =/= b so that they do not coincide.

These are x-coordinates to points on the graph of 5x + y = 7. But slope formula needs x-coordinates AND y-coordinates! To solve, we need to find the y-value by plugging in our x-value and solving for y. Then we have two points (a, y(a)) and (b, y(b))

At that point, we have our two general points. Then we must use slope formula to show for these two general points that
\( m = \dfrac{y(b) - y(a)}{b - a} = -5 \)
If all goes well, you should be able to simplify the left side to become the right side.