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Linear Systems

One of the most powerful applications of matrices is in solving linear systems: every linear system can be rewritten as a matrix equation, and then solved using the basic matrix operations we've already seen. For the larger systems that arise when studying real-world phenomenon, this matrix approach is much more effective than other ad-hoc techniques.

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 \(\left[\begin{array}{c} x\\ y\end{array}\right]?\)

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.

Consider the linear system \[\begin{aligned} 2x+y&=5,\\ x-3y&=1, \\x+y&=1.\end{aligned}\] Which of the following best describes the set of solutions to this system?

Consider the linear system \[\begin{aligned} 2x+y&=5,\\ x-3y&=1, \\3x-2y&=6.\end{aligned}\] Which of the following best describes the set of solutions to this system?

Consider the linear system \[\begin{aligned} 2x+y+z&=5,\\ x-3y+2z&=1.\end{aligned}\] Which of the following best describes the set of solutions to this system?

For what values of \(a\) will the following system of equations have no solutions? \[\begin{aligned} ax+2y&=4\\ 2x-ay&=6\end{aligned}.\]

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