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I have just started linear algebra (first week, we just finished going over gauss-jordan elimination) and am running into some difficulties in regards to a homework concept.

The stated problem:

"The curve $y = ax^2 + bc + c$ shown in the figure (I can't get this figure for you sorry!) passes through the points $(x_1, y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the coefficients $a, b,$ and $c$ are a solution of the system of linear equations whose augmented matrix is $$ \left[\begin{array}{rrrr} x_1^2 & x_1 & 1 & y_1 \\ x_2^3 & x_2 & 1 & y_2 \\ x_3^2 & x_3 & 1 & y_3 \end{array}\right] $$ "

I realize that the variables in this matrix are actually the a, b, c's and that the x's are in fact numbers, but I am really kind of lost on how to proceed with this type of problem. I am having a bit of a mental block on what is being shown to me here, and more so what I am aiming to do. We didn't cover anything like this in class and the next problem set has another similar problem in it, which I will briefly cover as well.

"Find the coefficients $a, b, c,$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation $y = ax^3 + bx^2 + cx + d$ " I am sorry I can't get the picture, if this is pivotal I can scan them in, there are 4 points given on the curve in the picture $(0,10), (1,7), (3,-11),$ and $(4,-14)$

Mostly I am wondering how to even approach these types of problem concepts and what I am supposed to be looking for? I hope I am making sense.

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You have stated that $y = ax^3 + bx^2 + cx + d$ and also that it is $y = ax^2 + bc + c$ - Which one is the correct one? – NoChance Jan 14 '12 at 20:53
Sorry, these are actually from two different questions, additionally my $ax^2 + bc + c$ should be $ax^2 + bx +c$ – vlc Jan 15 '12 at 21:41

I assume that when you write "the curve $y=ax^2+bc+c$" you actually mean $y=ax^2+bx+c$.

Now saying "the curve $y=ax^2+bx+c$ ... passes through the point $(x_1,y_1)$" simply says that the equation holds if you substitute the co-ordinates of the point in for the variables, that is, it just says $$y_1=ax_1^2+bx_1+c$$ Similarly for the other two points, so you get three equations for the unknowns $a,b,c$. Now, if someone hands you a system of three (linear) equations in three unknowns, do you know what is meant by the augmented matrix of that system? Do you know how to use the augmented matrix to solve the system?

EDIT in response to comment by OP: So let's forget about the particular problem of finding coefficients and look at the problem of solving a system of linear equations. Suppose you have a system of linear equations, in fact, suppose you have the system $$\eqalign{a+2b+3c&=4\cr2a+3b+4c&=5\cr3a+4b+6c&=1\cr}$$ The augmented matrix corresponding to this system is $$\pmatrix{1&2&3&4\cr2&3&4&5\cr3&4&6&1\cr}$$ Now you perform "row operations" on this matrix with the goal of bringing it to the form $$\pmatrix{1&0&0&q\cr0&1&0&r\cr0&0&1&s\cr}$$ which corresponds to the equations $$\eqalign{a&=q\cr b&=r\cr c&=s\cr}$$ which gives you the solution to your original system. So, what are these "row operations"? Well, you are allowed to interchange any two rows; you are allowed to multiply all the entries in any one row by any nonzero number, and you are allowed to subtract any multiple of one row from any other row. The details, along with many examples, will be found anywhere solving systems of linear equations is discussed, e.g., any Linear Algebra textbook, or search the web for "Gaussian elimination."

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I think I understand the points are substituted into the matrix, but how would you solve it and show that $a, b, c$ are solutions? I am not really sure what you mean by use the augmented matrix to solve the system. – vlc Jan 15 '12 at 21:45

solving $$ \left( \begin{array}{cccc} x_0^n&x_0^{n-1}&\dots&x_0&1\\ \dots&&&\\ x_n^n&x_n^{n-1}&\dots&x_n&1\\ \end{array} \right) \left( \begin{array}{c} a_n\\ \vdots\\ a_0\\ \end{array} \right) = \left( \begin{array}{c} y_0\\ \vdots\\ y_n\\ \end{array} \right) $$ for the coefficients $a_i$ gives the unique polynomial of degree $n$ going the $n+1$ points $(x_i,y_i)$ (assuming $x_i\neq x_j$). see vandermonde determinant

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