Question

What is the equation of the line passing through `(1,3), (4,6)`?

Answer 1

`y=x+2`

Explanation:

`"the equation of a line in "color(blue)"slope-intercept form"` is.

`•color(white)(x)y=mx+b`

`"where m is the slope and b the y-intercept"`

`"to calculate m use the "color(blue)"gradient formula"`

`•color(white)(x)m=(y_2-y_1)/(x_2-x_1)`

`"let "(x_1,y_1)=(1,3)" and "(x_2,y_2)=(4,6)`

`rArrm=(6-3)/(4-1)=3/3=1`

`rArry=x+blarrcolor(blue)"is the partial equation"`

`"to find b substitute either of the 2 given points into"`
`"the partial equation"`

`"using "(1,3)" then"`

`3=1+brArrb=3-1=2`

`rArry=x+2larrcolor(red)"is the equation of the line"`

Answer 2

`y=x+2`

Explanation:

First, we must know what an equation of a line looks like. We write the equation in slope-intercept form:

`y=mx+b`
(The `m` is the slope, and `b` is the y-intercept)

Next, find the slope (`m`) of the line by using the formula #(y_2-y_1)/ (x_2-x_1)#:

`((6)-(3))/((4)-(1))` `=` `3/3` `=` `1`

Next, find the y-intercept (`b`) by using the slope-intercept form equation and substituting `1` in for `m` and one of the ordered pairs in for `x` and `y`:

`(3)=(1)(1)+b` `->` `3=1+b` `->` `2=b`
-OR-
`(6)=(1)(4)+b` `->` `6=4+b` `->` `2=b`

Now, we can write the full equation of the line:

`y=x+2`
(We do not need to put a `1` in front of `x` because we know that `1` times any number equals itself)