Question

Find the equations of line joining points `(-2,3)` and `(1,4)`?

Answer 1

The equation for line joining two points `(x_1,y_1)` and `(x_2,y_2)` is given by `(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)` and for given points it is `y=1/3x+11/3`

Explanation:

Let the slope intercept form of equation be `y=mx+c`

here we do not know the slope `m` and `y`-intercept `c`

What we know is that this passes through the two coordinate pairs, say `(x_1,y_1)` and `(x_2,y_2)`.

As such we have three equations

`y=mx+c` ......(1)

`y_1=mx_1+c` ......(2) and

`y_2=mx_2+c` ......(3)

Now using these let us eliminate `m` and `c`

subtracting (2) from (1), we get `(y-y_1)=m(x-x_1)` ......(4)

and subtracting (2) from (3), we get `(y-2-y_1)=m(x_2-x_1)` ......(5)

Dividing (4) by (5)

`(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)`

As the two points are `(-2,3)` and `(1,4)`, the equation is

`(y-3)/(4-3)=(x-(-2))/(1-(-2))`

or `(y-3)/1=(x+2)/(1+2)`

or `y-3=(x+2)/3`

or `3y-9=x+2`

or `3y=x+11` or `y=1/3x+11/3`

Answer 2

For any point on the line, the coordinate pair in slope-intercept form is`( x, x/3+11/3)`

Explanation:

For slope m and intercept c, the equation is y = m x +c.

The slope intercept form for coordinates is (x, m x +c ).

The slope of the line through the given points is

`m = ( 4-3 ) / (1-(-2))=1/3`

Also, from (1, 4). 4 = i/3(1) + c. So, c = 11/3.

So, the answer is `( x, 1/3x+11/3)`.

Answer 3

The equation of the line is:

`y= 1/3x +11/3` which can be written as `y = 1/3x +3 2/3`

Explanation:

If you are given the coordinates of 2 points on a line, substituting them into the formula below allows you to find the equation immediately. In the process you also calculate the slope.

`(-2,3) and (1,4)`
`(x_1,y_1) and (x_2,y_2)`

`color(red)((y-y_1)/(x-x_1) = (y_2-y_1)/(x_2-x_1))" "larr RHS = m`

`(y-3)/(x-(-2))= (4-3)/(1-(-2)) = 1/3`

`(y-3)/(x+2) = 1/3" "larr` now cross-multiply

`3(y-3) = x+2color(white)(xxxxxxxxxx)or y-3 = 1/3(x+2)`

`3y -9 = x+2color(white)(xxxxxxxxxxxxxxxxx) y = 1/3x+2/3+3`

`3y = x+11color(white)(xxxxxxxxxxxxxxxxxxx) y = 1/3x+3 2/3`

`y= 1/3x +11/3`