## anonymous one year ago 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.

1. 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

2. 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 $$\LARGE \rm \color{blue}{m}$$

3. SolomonZelman

in this case: $$x_1=8$$ and $$y_2=3$$ The slope is $$\rm m$$

4. SolomonZelman

and in this case, the slope is 6 (i.e. m=6)

5. anonymous

So, if I understand what you're saying, the answer should be B instead of C?

6. 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. $$\text{slope between }(x,y)\text{ and }(x_0,y_0)=m\\\frac{y-y_0}{x-x_0}=m\\y-y_0=m(x-x_0)$$

7. SolomonZelman

Yes B is correct! VEry good!

8. 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

9. 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.

10. SolomonZelman

Yes, I think I can check more problems... and of course, N☼ PR☼BLEM

11. 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

12. anonymous

2. Graph the equation. y+5=-2(x-4)

13. anonymous

I'll attach the graphs.

14. 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: $$y-y_0=m(x-x_0)\\y-y_0=mx-mx_0\\y=\color{blue}mx+\underbrace{\color{red}{y_0-mx_0}}_{b}\\y=mx+b$$

15. SolomonZelman

if you got the graph why are you asked to graph it? :O

16. 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).

17. SolomonZelman

y+5=2(x-4) 1. expand the left side) 2. subtract 5 from both sides

18. anonymous

19. SolomonZelman

oh, -2.... I missed that. apologize.

20. SolomonZelman

ok, you can exclude option c, because it is going up and the line with a negative slope is always going down.

21. 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.

22. 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.)

23. anonymous

Possibly B?

24. 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

25. anonymous

Then it has to be A, because I don't think D is anywhere near what we're looking at.

26. SolomonZelman

yes

27. SolomonZelman

A is Correct

28. anonymous

Do you have time to help with a few more?

29. 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)

30. SolomonZelman

yes, I think so...

31. 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)

32. 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)

33. SolomonZelman

34. anonymous

35. anonymous

I think D would be the correct answer for that one.

36. SolomonZelman

Are you sure? I am asking that because the line is not going down, it si going up....

37. 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....

38. anonymous

OH! Okay! So A would be correct.

39. SolomonZelman

Yup

40. anonymous

41. SolomonZelman

you got "fooled" (excuse me) by the look of it.

42. SolomonZelman

|dw:1436469019837:dw|

43. anonymous

Possibly D?

44. 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)

45. 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?

46. anonymous

C, I believe.

47. 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

48. anonymous

Then it must be D.

49. SolomonZelman

it is B