Solving Parabolas & Lines Through (-1,3) and (2,12)

[DethRose]

I have a question assigned that states:

use gaussian elimination to find 2 distinct parabolas (y=ax^2+bx+c (one with a greater than 0 and one larger than 0)) that can pass through (-1,3) and (2,12). How many parabolas can pass through the given points and why?

Then use only your above results to determine the equation of the straight line that passes through the 2 points.


ok what i did is i made 2 equations:

3=a-b+c
12=4a+2b+c and used gaussian elimination to solve for the variables but the only one that can be found is c which equals -6.

Am i completely on the wrong track or is there something i am overlooking when I am doing this

please help

thanks

 

[TD]

Your equations look right but I don't think the result (c = -6) is correct, can you show us how you got that?

 

[DethRose]

1 1 1 3
2 2 1 12



can be transformed by a sequence of elementary row operations to the matrix

1 1 0 9
0 0 1 -6



Step 2: Interpret the reduced row echelon form


The reduced row echelon form of the augmented matrix is

1 1 0 9
0 0 1 -6



which corresponds to the system

1 x1 +1 x2 = 9
1 x3 = -6

 

[HallsofIvy]

I have a question assigned that states:
use gaussian elimination to find 2 distinct parabolas (y=ax^2+bx+c (one with a greater than 0 and one larger than 0)) that can pass through (-1,3) and (2,12). How many parabolas can pass through the given points and why?
Then use only your above results to determine the equation of the straight line that passes through the 2 points.
ok what i did is i made 2 equations:
3=a-b+c
12=4a+2b+c and used gaussian elimination to solve for the variables but the only one that can be found is c which equals -6.
Am i completely on the wrong track or is there something i am overlooking when I am doing this
please help
thanks

Because there are two equations, you should be able to determine two of the coefficients in terms of the third. That's why the problem says to find two such parabolas.
However, I also do not see how you got c= -6. Please show your work.

 

[DethRose]

1 -1 1 3
4 2 1 12



can be transformed by a sequence of elementary row operations to the matrix

1 0 1
--------------------------------------------------------------------------------
2 3
0 1 -1
--------------------------------------------------------------------------------
2 0



Step 2: Interpret the reduced row echelon form


The reduced row echelon form of the augmented matrix is

1 0 1
--------------------------------------------------------------------------------
2 3
0 1 -1
--------------------------------------------------------------------------------
2 0



which corresponds to the system

1 x1 +(1/2) x3 = 3
1 x2 +(-1/2) x3 = 0

x1 = +(-1/2) x3 +3
x2 = +(1/2) x3
x3 = arbitrary