Question

What equation represents the line that passes through (-8,11) and (4,7/2)

Answer 1

So our answers could be any of these depending on the form wanted*:

`y=(-5)/(8)x+6`

`5x+8y=48`

`y-11=(-5)/(8)(x+8)`

`y-(7)/(2)=(-5)/(8)(x-4)`

*There are other ways to write this equation.

Explanation:

So we are given two points on a line: (-8,11) and (4,7/2).

We can find the slope by using the formula
`(y_2-y_1)/(x_2-x_1) \text{ where } (x_1,y_1) \text{ and } (x_2,y+2) \text{ is on the line}`.

So to do this, I'm going to line up my points vertically and then subtract vertically, then put 2nd difference over 1st difference:

( 4 , 7/2)

-(-8 , 11)

----------------

12 -7.5

So the slope is -7.5/12 or -0.625 (If you type -7.5 division sign 12 in your calculator).

-0.625 as a fraction is -5/8 (just use the f<->d button to have your calculator convert your decimal to a fraction).

Anyways the equation of a line in slope-intercept form is y=mx+b where m is the slope and b is y-intercept.

We have m=-5/8 since that is the slope.

So plugging this into y=mx+b gives us y=(-5/8)x+b.

So now we need to find b. Pick one of the points given to you (just one).

Plug it into y=(-5/8)x+b and solve for b.

y=(-5/8)x +b with (-8,11)

11=(-5/8)(-8)+b

11=5+b

11-5=b

6=b

So the equation of the line in slope-intercept form is y=(-5/8)x+6.

We can also put in standard form which is ax+by=c where a,b,c are integers.

y=(-5/8)x+6

First step: We want to get rid of the fraction by multiplying both sides by 8:

8y=-5x+48

Second step: Add 5x on both sides:

5x+8y=48 (This is standard form.)

Now you can also out the line point-slope form,
`y-y_1=m(x-x_1) \text{ where } m \text{ is the slope and } (x_1,y_1) \text{ is a point on the line }`

So you can say either is correct:

`y-11=(-5)/(8)(x-(-8))`

or after simplifying:

`y-11=(-5)/(8)(x+8)`

Someone might have decided to use the other point; that is fine:

`y-(7)/(2)=(-5)/(8)(x-4)`

So our answers could be any of these depending on the form wanted*:

`y=(-5)/(8)x+6`

`5x+8y=48`

`y-11=(-5)/(8)(x+8)`

`y-(7)/(2)=(-5)/(8)(x-4)`

Answer 2

For this case we have that by definition, the equation of the line in slope-intersection form is given by:

`y = mx + b`

Where:

m: It's the slope

b: It is the cutoff point with the y axis

We have:

`(x1, y1): (- 8,11)\\(x2, y2): (4,3.5)`

`m = \frac {y2-y1} {x2-x1} = \frac {3.5-11} {4 - (- 8)} = \frac {-7.5} {4 + 8} = \frac {-7.5} {12 } = - \frac {\frac {15} {2}} {12} = - \frac {15} {24} = - \frac {5} {8}`

Thus, the equation will be given by:

`y = - \frac {5} {8} x + b`

We substitute a point to find "b":

`11 = - \frac {5} {8} (- 8) + b\\11 = 5 + b\\b = 11-5\\b = 6`

Finally:

`y = - \frac {5} {8} x + 6`

Answer:

`y = - \frac {5} {8} x + 6`