f(5)=5 and f(7)=-4 find the equation for f

Question

I am unable to figure this problem I have tried several times  thank you

Answer 1

Two unique points define a line.  Other geometric figures also may pass through those two points (you will learn those later).

There are several forms of the equation of a line:
      Standard form                           Ax + By = C
      Slope-intercept form                  y = mx + b
      Point-slope form                        y – y₁ = m(x – x₁)
      Two-point form              y – y₁ = [ (y₂ – y₁)/(x₂ – x₁) ](x – x₁)

O.K., you picked the two-point form, right?   
                The points are  P₁:  (5,5) and P₂: (7,-4)

     y - 5 = [ ((-4) - 5)/(7 - 5) ] (x - 5)          (put in x and y values of points)
     y - 5 = [  -9 / 2 ](x - 5)                         (simplify)
     y - 5  =  -9/2x  + 45/2                           (distribute -9/2)
      y =  f(x) = -9/2x + 55/2                   (add 5 to both sided)

Check (very important):
   Is (5,5) on the line?      5 = (-9/2)(5) +  55/2  ?
                                          5 =  -45/2   +  55/2  ?
                                           5 =  10/2         yes.

   Is (7,-4) on the line?     -4 = (-9/2)(7) + 55/2       ?
                                           -4  =  -63/2   + 55/2   ?
                                           -4  =   -8/2   ?       yes.

Answer 2

Is this a linear equation?  If so, its solution is as follows: its a line passing through two points (x₁, f(x₁)) and (x₂,f(x₂)), or (5,5) and (7,-4). the slope m is found from

 

m = (y₂ - y1)/ (x₂ - x₁) = (-4-5)/(7-5) = -9/2.

 

now that we know the slope, we can use the point slope form of the equation of a line,

 

y-y₁ = m(x-x₁), to find our equation.

 

either point will do. Use the first one.

 

y-5 = -9/2 (x-5)  distribute the -9/2 on teh right side

y-5 = -9x/2 +45/2  adding 5 (= 10/2) to both sides

y =  -9x/2 +55/2  Our answer in slope=intercept form

 

or, putting both x and y terms on the left side,

 

9x/2 + y = 55/2 and multiplying both sides by the least common denominator,2, we get

 

9x + 2y = 55 in standard form.