CPhill

Yeah...substitution can be easier  a lot of the time...

See my answer, here  :

https://web2.0calc.com/questions/mathematics_38

OK....you're off to a good start  !!!....most people have difficulty even setting up the equations  !!!

Let V  be the number of students a  van can carry and B be the number of students that a bus can carry....you have the correct equations...we just need to solve them..we can use substitution.

2V + 12B  = 520        divide through by 2  ⇒   V  + 6B  = 260  ⇒  V = 260 - 6B    (1)

13V + 3B  =  230   (2)

Put  (1) into (2)  for V  and we have

13 ( 260 - 6B)  +  3B  = 230    simplify

3380 - 78B + 3B  = 230

3380  - 75B   = 230    subtract 3380  from each side

-75B = -3150   divide both sides by  -75

B  = 42    = the students a bus can carry

And using (1)  we have that  V  = 260 - 6(42)   = 8  =  the number of students that a van can carry...!!!! 

What didn't work  ??

(-4, 9)...   10  left  7 down  puts her at  (  -4 - 10 , 9  - 7)  = ( -14, 2)

So....  the points  ( -4, 9)  and (-14,2)  are on the  line

Slope of this line  =   [ 2 - 9] / [ -14 - -4 ]  =  [ -7] / [ -10]  =  7/10

Equation of the line

y  = (7/10)(x - - 4)  + 9

y  = (7/10) ( x + 4)  + 9     multiply through by 10

10y = 7 (x + 4)  + 90

10y = 7x + 28  + 90

-7x + 10y  = 118

Here's a graph : https://www.desmos.com/calculator/jlpnh4ai3p

2x + 7  = -5      

2x  = -12

x = -6

-x^2 - x + 1    = -5     multiply through by -1

x^2 + x - 1  = 5

x^2 + x  - 6  = 0   factor

(x + 3) ( x - 2)  =  0

Set each factor to  0  and solve for  x and we get   x  = -3   (out of the domain for the function)  or  x = 2

So...the sum of the  x values such that f(x)   = -5  ⇒   -6  + 2   =  -4 

2, 4 , 6 , 8, 10

Yep...taking turns driving is the way I interpret  it...each drives 12 hours

So each drove  12 hrs

And Rate * Time = Distance

So...call Vlado's rate R  and Santi's rate (.75R)    and we have

Santi's Rate * Time Driving  + Vlado's Rate  * Time Driving  = Total Distance

.75R (12)   + R(12)  = 1304  simplify

R(9) + R(12)  =  1304

R (21)  = 1304   divide both sids by 21

R ≈ 62.1  mph

So Santi  drove  ≈   .75 (62.1) (12)  ≈ 559 miles ⇒  round to   560 miles

See  under my original answer ...it won't change anything, much...

Here's a corrected graph : https://www.desmos.com/calculator/rqfmbx0gfb