Question 822037
You know equations representing a line that can be written as
{{{y=mx+b}}} with some constants {{{m}}} and {{{b}}} .
You can think of those equations as linear functions with {{{y}}} being a function of {{{x}}} .
You could write them as {{{f(x)=mx+b}}} .
That is the type of equation you need to find.
There's more than one way to the answer.
You could substitute the values given and solve the resulting system of equations for {{{m}}} and {{{b}}} .
You could also draw on what you know of equations of a line,
figure out the slope,
use one point to write the equation in point-slope form,
and then solve for {{{y}}} to get the slope-intercept form of the equation.
 
SOLVING A SYSTEM:
We substitute {{{f(6)=-4}}} and {{{f(9)=-9}}} into {{{f(x)=mx+b}}} to get two equations.
{{{system(-4=6m+b,-9=9m+b)}}}
Subtracting the first equation from the second, we get
{{{-9+4=9m-6m}}} --> {{{-5=3m}}} --> {{{m=-5/3}}}
Substituting that value for {{{m}}} into the first equation, we get
{{{-4=6*(-5/3)+b}}} --> {{{-4=-10+b}}} --> {{{-4+10=b}}} --> {{{b=6}}}
So the linear function is {{{highlight(f(x)=(-5/3)x+6)}}}
 
FINDING THE EQUATION OF A LINE:
The information given, tells you that
for {{{x=6}}} , {{{y=-4}}}  so point (6,-4) is on the line,
and it also tells you that
for {{{x=9}}} , {{{y=-9}}}  so point (9,-9) is on the line.
The slope is the difference in {{{y}}} values divided by the difference in {{{x}}} values:
{{{m=(-9-(-4))/(9-6)=(-9+4)/3=-5/3}}}
With that slope and knowing that the line goes through (6,-4) we can write a point-slope form of the equation as
{{{y-(-4)=(-5/3)(x-6)}}} --> {{{y+4=(-5/3)(x-6)}}}
We can solve for {{{y}} to get the slope-intercept form:
{{{y+4=(-5/3)(x-6)}}} --> {{{y+4=(-5/3)x-6(-5/3)}}} --> {{{y+4=(-5/3)x+10}}} --> {{{y=(-5/3)x+10-4}}} --> {{{y=(-5/3)x+6}}}
Then you would write it with {{{f(x)}}} instead of {{{y}}} , as
{{{highlight(f(x)=(-5/3)x+6)}}}