Question

What is the equation of the line with slope `m= 1` that passes through `(7,13)`?

Answer 1

`(y-13)=1(x-7)` in explicit slope-point form
or
`x-y=-6` in standard form

Explanation:

General slope-point form is
`color(white)("XXX")(y-bary)=m(x-barx)`
for a line with slope `m` through the point `(barx,bary)`

Substituting the given values results in
`color(white)("XXX")y-13=1(x-7)`

To convert to standard form: `Ax+By=C`
`color(white)("XXX")-x+y-13=-7`

`color(white)("XXX")-x+y=6`

`color(white)("XXX")x-y=-6`

or, in explicit standard form:
`color(white)("XXX")1x-1y=-6`

Answer 2

A different approach to solving this one!

`y=x+6`

Explained with a lot of detail!

Explanation:

`color(blue)("Assumption "-> " this is a strait line graph")`

Standard equation form for a strait line graph:

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

Given: slop (gradient) `->m=1`

so equation (1) becomes:

`y=(1)x+c`

But `1xx x =x`

`y=x+c`........................(2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("To find the value of "c)`
Using given point of `x=7 and y=13`

Equation (2) becomes:

`y=x+c color(white)(..x.)->color(white)(.x..)13=7+c`

`color(blue)(c=13-7= 6)` as predicted.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`y=x+6`