Question

What is the equation of the line that passes through the points (4/5,1/5) and (1/2,3/2)?​

Answer 1

`y-(3)/(2)=(-13)/(3)(x-(1)/(2))` point-slope form

`13x+3y=11` (standard form)

Let me know if you prefer another form.

Explanation:

The slope of a line can be found using
`(y_2-y_1)/(x_2-x_1)` provided you are given two points on the line.

We are.

Now you can use that formula. But I really love to just line up the points vertically then subtract them vertically then put 2nd difference over 1st difference.

(4/5 , 1/5)

-( 1/2 , 3/2)

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

3/10 -13/10

2nd/1st =
`((-13)/(10))/((3)/(10))=(-13)/(3)` is our slope.

So the following is point-slope form for a linear equaiton:

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

Plug in a point
`(x_1,y_1)=((1)/(2),(3)/(2)) \text{ and } m=(-13)/(3)`.

This gives:

`y-(3)/(2)=(-13)/(3)(x-(1)/(2))`

I'm going to distribute:

`y-(3)/(2)=(-13)/(3)x-(-13)/(6)`

Now I don't like these fractions so I'm going to multiply both sides by the least common multiply of 2,3, and 6 which is 6:

`6y-9=-26x+13`

Add 26x on both sides:

`26x+6y-9=13`

Add 9 on both sides:

`26x+6y=22` This is actually standard form of a line.

It can be simplified though.

Divide both sides by 2:

`13x+3y=11` (standard form)

Answer 2

`\large\boxed{y=-(13)/(3)x+(11)/(3)}-\bold{slope\ intercept\ form}\\\boxed{13x+3y=11}-\bold{standard\ form}`

Explanation:

The slope-intercept form of an equation of a line:

`y=mx+b`

m - slope

b - y-intercept

The formula of a slope:

`m=(y_2-y_1)/(x_2-x_1)`

We have two points

`\left((4)/(5),\ (1)/(5)\right),\ \left((1)/(2),\ (3)/(2)\right)`

Convert fractions to the decimals

(divide the numerator by the denominator) :

`(4)/(5)=0.8,\ (1)/(5)=0.2,\ (1)/(2)=0.5,\ (3)/(2)=1.5`

`\left((4)/(5),\ (1)/(5)\right)=(0.8,\ 0.2)\\\\\left((1)/(2),\ (3)/(2)\right)=(0.5,\ 1.5)`

Calculate the slope:

`m=(1.5-0.2)/(0.5-0.8)=(1.3)/(-0.3)=-(13)/(3)`

Put the value of slope and the coordinates of the first point to the equation of a line:

`0.2=-(13)/(3)(0.8)+b` multiply both sides by 3

`0.6=(-13)(0.8)+3b`

`0.6=-10.4+3b` add 10.4 to both sides

`11=3b` divide both sides by 3

`(11)/(3)=b\to b=(11)/(3)`

Finally:

`y=-(13)/(3)x+(11)/(3)` - slope-intercept form

Convert to the standard form (Ax + By = C):

`y=-(13)/(3)x+(11)/(3)` multiply both sides by 3

`3y=-13x+11` add 13x to both sides

`13x+3y=11` - standard form