Question

correct??

The graph of a line passes through two given points, as shown on the graph below. What is the equation of the line written in general form?



https://media.glynlyon.com/g_alg01_2013/3/77c.gif
x - y + 5 = 0 <--
x + y - 5 = 0
x - y - 5= 0

Answer 1

The points are, let's name them a(0,5) and b(5,0)
we have a very interesting structure, wich I won't prove, just write it here, that says the following:
\[(y-y _{1})=m(x-x _{1})\]
that is the structure of the line that goes by one point, but for that we have to know the slope, and conveniently, for the slope, we have two points:
\[m=\frac{ y _{b}-y _{a} }{ x _{b}-x _{a} }\]
Replacing and dividing we get that m= -1
and replacing in the first formula we get a equation of:
\[(y-0)=-1(x-5)\]
operating a little:
\[y=-x+5\]
since the answers are all in the general formula of the line, might as well convert the one I just got into it's general formula:
\[y+x-5=0\]
by conmutative of the sum I can switch places with x and y:
\[x+y-5=0\]
So the correct answer was the second formula and not the first.

Answer 2

@Owlcoffee thank you so much for your amazing explanation! do you mind correcting me if i got this question right??

The slope of a line is -, and the y-intercept is 5. What is the equation of the line written in general form?

x - 5y - 25 = 0
x - 5y - 5 = 0
x + 5y - 25 = 0 <---!

Answer 3

Now for taht exercise we'll recall the point-slope formula, it has this structure:
\[y=mx+b\]
where m is the slope and b the y-intercept.
in the problem, they tell us that the slope is "-" wich is also traduced as "-1" and the y intercept is 5, meaning that the line equation will look like this:
\[y=-x+5\]
the general formula means that everything has to be equal zero, so I'll express that line in general formula, but since the terms are proportional, and the slope always remain constant and I only pivot the numbers, I can say that:
\[5y=-x+25\]
expressing in general formula:
\[x+5y-25=0\]
So you are correct.

Answer 4

eq in intercept form is
\[\frac{ x }{5 }+\frac{ y }{5 }=1,x+y=5,x+y-5=0\]

Answer 5

That's true, thing is, they pivoted the points in a proportional way, and since the slope is pretty much constant you will get the third equation as a result of that pivoting.

Answer 6

here x and y intercepts are positive and each equal to 5.
General eq. in intercept form is
\[\frac{ x }{a }+\frac{ y }{b }=1,a \neq0,b \neq0 a,b \in R\]