Write an equation for a line passing through the points ( 1 , 6 ) and ( − 2 , 0 )

[Gadsilla]

Write an equation for a line passing through the points ( 1 , 6 ) and ( − 2 , 0 )

Write an equation for a line passing through the points (1,6) and (−2,0)

y=mx+b

6=mx+b
0=mx+b

6-0 = 6

6=mx+b

Now how do I proceed ?

6 =m(1)+b - m(-2)+b
(1)(-2)=-2

y=m(-2)+b

(6)-(-2)=8

y=-2x+8

Is what I did, but it's wrong.

 

[mmm4444bot]

Determine the slope first. There's a formula for that. It's called the slope formula, and it uses the coordinates from the two given points. (Click the link, study some lessons.)

Once you know the slope, you have two ways to proceed:

(1) write the slope-intercept form: y = mx + b (substituting your slope value for m), and then substitute coordinates from one point for x and y, to obtain an equation containing only symbol b; solve for b

(2) substitute the slope and the coordinates of one point into the point-slope formula

Let's leave method (2) for later.


EG (1): Write an equation for the line passing through (0,2) and (3,7)

Use the slope formula: m = (7-2)/(3-0) = 5/3

Write y=mx+b:

y = (5/3)x + b

Pick either point and substitute for x and y, then solve for b:

7 = (5/3)(3) + b

7 = 5 + b

b = 2

Answer: y = (5/3)x + 2

Use the other given point, to check this answer:

y = (5/3)x + 2

2 = (5/3)(0) + 2

2 = 2

Yay

 

[Gadsilla]

Determine the slope first. There's a formula for that. It's called the slope formula, and it uses the coordinates from the two given points. (Click the link, study some lessons.)

Once you know the slope, you have two ways to proceed:

(1) substitute the slope and the coordinates of one point into the point-slope formula

(2) write (slope-intercept form) y = mx + b (using your slope value for m) and then substitute coordinates from one point for x and y, to solve for b

Let's leave method (2) for later.


EG (1): Write an equation for the line passing through (0,2) and (3,7)

Use the slope formula: m = (7-2)/(3-0) = 5/3

Write y=mx+b:

y = (5/3)x + b

Pick either point and substitute for x and y, then solve for b:

7 = (5/3)(3) + b

7 = 5 + b

b = 2

Answer: y = (5/3)x + 2

Use the other given point, to check this answer:

y = (5/3)x + 2

2 = (5/3)(0) + 2

2 = 2

Yay

Could you show me how you'd do it and explain the reasoning behind the steps for finding the slope. I understand best that way.
So It's basicly (x1-x2)/(y1-y2) = slope

(1-(-2))/(6-0)=0.5

0.5 = 1/2

Slope = 1/2

Now what ?

 

[HallsofIvy]

Write an equation for a line passing through the points (1,6) and (−2,0)

y=mx+b

6=mx+b
0=mx+b

These are not true. y= 6 only for x= 1 not for any other x. Since you have replaced y with 6 you must replace x with 1. Similarly y= 0 only for x= -2. You should have
6= m(1)+ b and
0= m(-2)+ b

So you have the two equations m+ b= 6 and -2m+ b= 0 to solve for m and b.

6-0 = 6

6=mx+b

Now how do I proceed ?

6 =m(1)+b - m(-2)+b
(1)(-2)=-2

y=m(-2)+b

(6)-(-2)=8

y=-2x+8

Is what I did, but it's wrong.

 

[JeffM]

Could you show me how you'd do it and explain the reasoning behind the steps for finding the slope. I understand best that way.
So It's basicly (x1-x2)/(y1-y2) = slope

THIS IS INCORRECT. This is the reciprocal of the slope. See below for more detail.

(1-(-2))/(6-0)=0.5

0.5 = 1/2

Slope = 1/2

Now what ?

Let's think function notation for a moment. (In a response to a previous question of yours, I explained that the slope is a basic descriptor for "smooth" functions.)

We usually say y = f(x). That is purely conventional; we could say x = f(y). But let's stick with the usual convention. When we say y = f(x), we mean that we shall be able to determine y once we know x. We say y is the dependent variable and x is an independent cariable. (It is not generally true that you can determine an independent variable just from knowing the dependent variable although that is true for some functions.)

So what does the slope tell us. Loosely, it gives the ratio of the amount of change in the dependent variable (y) compared to the amount of change in an independent variable (x). Because the change in y may possibly be zero, we cannot safely divide by the amount of change in y. But we are assuming some degree of change in x and can be safely assured that the change in x is not zero.

\(\displaystyle \text {slope } \approx \dfrac{\text {change in y}}{\text {change in x}}.\)

The formula above is exact for a linear function. In other words, the slope of the linear function y = f(x) = mx + b is

\(\displaystyle \dfrac{y_1 - y_2}{x_1 - x_2} = \dfrac{(mx_1 + b) - (mx_2 + b)}{x_1 - x_2} = \dfrac{mx_1 - mx_2}{x_1 - x_2} = \dfrac{m(x_1 - x_2)}{(x_1 - x_2)} = m.\)

 

[Steven G]

Write an equation for a line passing through the points (1,6) and (−2,0)

y=mx+b

6=mx+b
0=mx+b

6-0 = 6

6=mx+b

Now how do I proceed ?

6 =m(1)+b - m(-2)+b
(1)(-2)=-2

y=m(-2)+b

(6)-(-2)=8

y=-2x+8

Is what I did, but it's wrong.

y=mx+b OK

6=mx+b x should be 1, ie 6= 1x+b
0=mx+b x should be -2, ie 6= -2x+b

6-0 = 6 OK

6=mx+b No, not at all. I agree that 6-0=6, but YOU should be computing (mx+b)-(mx+b) and getting zero - NOT mx+b

 

[mmm4444bot]

… [could you] explain the reasoning behind the steps for finding the slope. I understand best that way.

I could, yes. But you've already seen explanations with the reasoning behind the definition of slope and the steps to determine it. If you didn't understand those, why would mine be different?

How many lessons (videos and written combined) did you study, from the link that I provided? How much time did you spend on that? Did you work through their examples, writing down each step on paper? Was there anything specific presented that you did not understand? Were you able to follow the example exercise, in post #2? Did you use paper and pencil for that? Is there anything in that post you don't understand?

So It's basicly (x1-x2)/(y1-y2) = slope

I'm sorry; it's not. Were you to have worked though several examples, having followed a number of different presentations of slope's meaning and the formula (from different authors), I doubt you would have made this mistake. Please go back and try again. Once you're able to post the correct slope formula, follow the steps in post #2. Show your corrected work. We can go from there. :cool:

Once you know the slope … write the slope-intercept form: y = mx + b (substituting your slope value for m), and then substitute coordinates from one point for x and y, to obtain an equation containing only symbol b; solve for b …