# Our first dip into a new way of thinking

Now let’s take the same two equations as last time, but view them in a slightly different way. Call this the column view of the system. $x\underbrace{ \begin{bmatrix}2 \\ -1 \end{bmatrix}}_\mathrm{first \ vector} +\ y \underbrace{ \begin{bmatrix}-1 \\ 2 \end{bmatrix}}_\mathrm{second \ vector} = \underbrace{ \begin{bmatrix}0 \\ 3 \end{bmatrix}}_{resultant \ \mathrm{vector}}$

Notice that we have placed the coefficients of each variable into a 2 x 1 matrix. The constant terms also appear in a 2 x 1 matrix. If we think of the matrices as vectors, we can restate the solution of the system of equations in another way:

“How can we combine the vectors of the left hand side of the equation, in just the right amounts, to get the vector on the right hand side of the equation? This right amount is called the linear combination that solves the original system of equations. The two vectors on the left hand side can be thought of as the ingredients of this particular problem. The solution is the mix of these ingredients that solves the problem.

If we take 1 of the first vector (i.e.: x = 1) and 2 of the second vector (y = 2), we get this: $\begin{bmatrix}2 \\ -1 \end{bmatrix} + \begin{bmatrix}-2 \\ 4 \end{bmatrix} = \begin{bmatrix}0 \\ 3 \end{bmatrix}$

In other words, the solution is $x = 1, y = 2$ or $(1, 2)$. The same solution we obtained using the Gauss Method!

We did this approach graphically on the board in class.

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