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Consider the linear system \[\begin{aligned} 2x+y&=5,\\ x-3y&=1.\end{aligned}\] This can be written in matrix form as

\[\left[\begin{array}{cc} 2 & 1\\ 1 & -3 \end{array}\right]\left[\begin{array}{c} x\\ y\end{array}\right]=\left[\begin{array}{c} 5\\ 1\end{array}\right].\]

If \(M=\left[\begin{array}{cc} 2 & 1\\ 1 & -3 \end{array}\right],\) which of the following is the correct expression for the vector \(\left[\begin{array}{c} x\\ y\end{array}\right]?\)

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The previous example highlights the nicest case of linear systems: when we have as the same number of equations as variables, and a single solution.

In practice though, we often run into systems with more equations than variables or vice versa. Understanding the solution sets of these systems is crucial to using them as models.

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