# Thread: Proof with System of Linear Equations

1. ## Proof with System of Linear Equations

The curve $\displaystyle y=ax^2 + bx + c$ passes through the points $\displaystyle (x_1, y_1), (x_2, y_2), (x_3, y_3)$. Show that the coefficients a, b, and c are a solution of the system of linear equations whose augmented matrix is:

$\displaystyle \begin{bmatrix} x^2_1& x_1& 1& y_1 \\ x^2_2& x_2& 1& y_2 \\ x^2_3& x_3& 1& y_3 \end{bmatrix}$

Just began linear algebra and having trouble understanding what this question is asking. I thought only constants/coefficients could go in the augmented matrix, but the x's and y's are variables. Confusing.

2. Can you write out three equations that you know the curve satisfies? I would then compare those three equations with matrix multiplication, and see if the result pops out at you.

What goes into a matrix can be all sorts of things, depending on the context.

3. Technically, what's in that matrix are all constants.

4. Originally Posted by Ackbeet
Can you write out three equations that you know the curve satisfies? I would then compare those three equations with matrix multiplication, and see if the result pops out at you.

What goes into a matrix can be all sorts of things, depending on the context.
I think the three equations are:
$\displaystyle ax^2_1 + bx_1 + c = y_1$
$\displaystyle ax^2_2 + bx_2 + c = y_2$
$\displaystyle ax^2_3 + bx_3 + c = y_3$
But we have not covered matrix multiplication yet. That is the next section of the book. Is there anyway to do this without matrix multiplication?

5. Well, if you haven't done matrix multiplication yet, you must have some method of constructing the augmented matrix. In fact, you can read off the coefficients of the augmented matrix from the three (incidentally correct) equations you wrote out. Does that make sense?

6. Originally Posted by Ackbeet
Well, if you haven't done matrix multiplication yet, you must have some method of constructing the augmented matrix. In fact, you can read off the coefficients of the augmented matrix from the three (incidentally correct) equations you wrote out. Does that make sense?
Yes, I think it's making sense now. The most confusing thing was accepting that $\displaystyle x_1$, etc. were acting as the constants/coefficients.
Thanks!

7. You're very welcome. Have a good one!