Math Help - 2 short problems

1. 2 short problems

Hi, it's been awhile since I've been on here but I've now officially started Linear Algebra.

There are two problems in my homework I cannot solve. They are the following:

1. Find the equation of the form $y = mx + b$ that passes through points $(4, -1)$ and $(1,8)$ using elementary row operations on a matrix.

2. Find the equation of the parabola $y = ax^2 + bx + c$ that pass through (0,3), (1,9) and (2,3) using elementary row operations on a matrix.

I believe I have the second problem solved but I am unsure of my answer. Thank you for looking at this! =)

2. Show us how you've done the second one and we'll comment.

3. Actually, I think I may be on my way to answering them both. It's just that I don't know what to do when I get to a certain point. Here goes:

for the y = mx + b problem....

I set it up as $-1 = m(4) + b$ and $8 = m(1) + b$.

I set up a matrix that looks like:

4 1 -1
1 1 8

I then set it up in rref form using about 8 steps, until I come out to a matrix that looks like...

1 0 -3
0 1 11

However, I don't really know what this tells me. I don't know how to check it, or set it up in an equation form. I get to the same point in the quadratic problem.

4. Originally Posted by TheBerkeleyBoss
Actually, I think I may be on my way to answering them both. It's just that I don't know what to do when I get to a certain point. Here goes:

for the y = mx + b problem....

I set it up as $-1 = m(4) + b$ and $8 = m(1) + b$.

I set up a matrix that looks like:

4 1 -1
1 1 8

I then set it up in rref form using about 8 steps, until I come out to a matrix that looks like...

1 0 -3
0 1 11

However, I don't really know what this tells me. I don't know how to check it, or set it up in an equation form. I get to the same point in the quadratic problem.
The whole point of row reduction is that you have transformed the equations 4m+ b= -1 and m+ b= 8 into 1m+ 0b= m= -3 and 0m+ 1b= b= 11. Now, what do you think the equation y= mx+ b is?

5. Thank you HallsOfIvy. I was able to figure it out from there. I want to understand why this works. Why exactly does rref form work the way it does? I really want to conceptually understand this. What is going on at a geometric level?