anonymous
  • anonymous
Can someone check my answers please? I'm really struggling with this. Write an equation in point-slope form for the line through the given point with the given slope.
Algebra
chestercat
  • chestercat
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anonymous
  • anonymous
(8,3); m=6 (1 point) y+3=6(x-8) y-3=6(x-8) y-3=6(x+8) <-- My answer. y+3=6x+8
SolomonZelman
  • SolomonZelman
\(\LARGE y-\color{green}{y_1}=\color{blue}{\rm m}(x-\color{red}{x_1})\) where your point is \(\LARGE \left(\color{red}{x_1},\color{green}{y_1}\right)\) and your slope is \(\LARGE \rm \color{blue}{m}\)
SolomonZelman
  • SolomonZelman
in this case: \(x_1=8\) and \(y_2=3\) The slope is \(\rm m\)

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SolomonZelman
  • SolomonZelman
and in this case, the slope is 6 (i.e. m=6)
anonymous
  • anonymous
So, if I understand what you're saying, the answer should be B instead of C?
anonymous
  • anonymous
if a line has a slope \(m\) and passes through a point \(x_0,y_0\) then the slope between any other point on the line \((x,y)\) and the given point \((x_0,y_0)\) must be \(m\), i.e. $$\text{slope between }(x,y)\text{ and }(x_0,y_0)=m\\\frac{y-y_0}{x-x_0}=m\\y-y_0=m(x-x_0)$$
SolomonZelman
  • SolomonZelman
Yes B is correct! VEry good!
anonymous
  • anonymous
so in this case our given point is \((8,3)\) and our slope is \(6\). for any other point \((x,y)\) on our line we require that the slope between \((x,y)\) and \((8,3)\) is \(6\), so:$$\frac{y-3}{x-8}=6\\\text{so multiplying both sides by }x-8\text{ gives us: }y-3=6(x-8)$$... and this is the point-slope form
anonymous
  • anonymous
Thank you, would you mind checking a few more? I have two days to finish and I'm terrified of getting a bad grade.
SolomonZelman
  • SolomonZelman
Yes, I think I can check more problems... and of course, N☼ PR☼BLEM
anonymous
  • anonymous
now this is an unrelated, tangential note to help connect this with slope-intercept form: suppose the given point is a \(y\)-intercept, i.e. a point whose \(x\)-coordinate is zero; we can write this \((0,b)\). if we're given this \(y\)-interept and a slope \(m\), we'll find the following equation by point-slope form:$$y-b=m(x-0)$$simplifying since \(x-0=x\) we have:$$y-b=mx\\y=mx+b$$which is the normal slope-intercept form that you're familiar with
anonymous
  • anonymous
2. Graph the equation. y+5=-2(x-4)
anonymous
  • anonymous
I'll attach the graphs.
anonymous
  • anonymous
unrelated note continued: in fact, given a point-slope form equation \(y-y_0=m(x-x_0)\) there is a natural way to rewrite in slope-intercept form -- simply distribute everything out and get \(y\) alone on one side: $$y-y_0=m(x-x_0)\\y-y_0=mx-mx_0\\y=\color{blue}mx+\underbrace{\color{red}{y_0-mx_0}}_{b}\\y=mx+b$$
SolomonZelman
  • SolomonZelman
if you got the graph why are you asked to graph it? :O
SolomonZelman
  • SolomonZelman
It will be easier if we go ahead and simplify this equation written in a point slope form, INTO, a y-intercept form (i.e. y=mx+b).
SolomonZelman
  • SolomonZelman
y+5=2(x-4) 1. expand the left side) 2. subtract 5 from both sides
anonymous
  • anonymous
SolomonZelman
  • SolomonZelman
oh, -2.... I missed that. apologize.
SolomonZelman
  • SolomonZelman
ok, you can exclude option c, because it is going up and the line with a negative slope is always going down.
SolomonZelman
  • SolomonZelman
y+5=-2(x-4) you can re-write it into a s y-intercept form, but you don't really have to. you know the line should go down by 2 units every time it goes 1 unit to the right (that is what a slope of -2 means). What you don't know is where do you start.... plug in x=0, to find the y-interecept.
SolomonZelman
  • SolomonZelman
(I mean I can tell a negative slope that is -2, and a negative slope that is not as steep as -2 right away, tho' so you can just look at the graph to see the option (the only one option) that goes 2 units down every time it goes 1 unit to the right.)
anonymous
  • anonymous
Possibly B?
SolomonZelman
  • SolomonZelman
B is more like going 2 units to the right as it goes 1 down, but a slope of -2 means that we go down by 2 each time we go 1 to the right
anonymous
  • anonymous
Then it has to be A, because I don't think D is anywhere near what we're looking at.
SolomonZelman
  • SolomonZelman
yes
SolomonZelman
  • SolomonZelman
A is Correct
anonymous
  • anonymous
Do you have time to help with a few more?
SolomonZelman
  • SolomonZelman
(I am just showing an SAT kind of a technique where you can quickly identify the answer and exclude other options just using a quick look/analysis)
SolomonZelman
  • SolomonZelman
yes, I think so...
anonymous
  • anonymous
Which point is located on the line represented by the equation y+4=-5(x-3)? (-4,-5) (-5,-4) (3,-4) <-- My answer. (-3,4)
SolomonZelman
  • SolomonZelman
Recall the point slope formula rule. I will rewrite your equation for you real quick. y-(-4)=-5(x-3) now compare that to y-y1=m(x-x1)
SolomonZelman
  • SolomonZelman
Yes, your answer is right
anonymous
  • anonymous
1 Attachment
anonymous
  • anonymous
I think D would be the correct answer for that one.
SolomonZelman
  • SolomonZelman
Are you sure? I am asking that because the line is not going down, it si going up....
SolomonZelman
  • SolomonZelman
It does go by 2 units vertically, as it goes 1 unit to the right. BUT it goes 2 units (vertically) up, not (vertically) down....
anonymous
  • anonymous
OH! Okay! So A would be correct.
SolomonZelman
  • SolomonZelman
Yup
anonymous
  • anonymous
I'm really hoping that I'm right about this one, but I think it's A.
1 Attachment
SolomonZelman
  • SolomonZelman
you got "fooled" (excuse me) by the look of it.
SolomonZelman
  • SolomonZelman
|dw:1436469019837:dw|
anonymous
  • anonymous
Possibly D?
SolomonZelman
  • SolomonZelman
see it perfectly goes 3 units down, as it goes 8 units to the right (and although it is close to the slope of 1/2)
SolomonZelman
  • SolomonZelman
-8/3 slope, means: Your graph goes 8 units down as it goes 3 units to the right -3/8 slope, means: Your graph goes 3 units down as it goes 8 units to the right which one is correct if you look on our graph?
anonymous
  • anonymous
C, I believe.
SolomonZelman
  • SolomonZelman
it wuld be C if it was going UP by 3 while going 8 to the right. But it goes DOWN by 3 while it goes 8 to the right. So m=-3/8
anonymous
  • anonymous
Then it must be D.
SolomonZelman
  • SolomonZelman
it is B

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