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Section 1.5 Systems of Linear Equations

Finding the coefficients needed to describe a given point as a linear combination of other points often requires solving serveral linear equations simultaneously.

Definition 1.5.1.

A linear equation is an equation of the form:

\begin{equation*} a_1 x + a_2 y + a_3 z=b \end{equation*}

where the \(a_i\) represent specific real numbers called constants, \(b\) is also a real number and \(x\text{,}\) \(y\) and \(z\) are variables or unknowns. The solution to a linear equation is the set of ordered triples of real numbers, \(\ot{x}{y}{z}\) for which the equation is true.

It is often important to be able to find solutions that work for multiple linear equations. To solve such a system of linear equations, transform it into simpler and simpler systems whose solutions are identical to the given system. Then use the simplified system to describe the values of \(x\text{,}\) \(y\) and \(z\) that make all equations in the system true, and give the result using set notation: \(\mathbb{S}= \{\ot{\hspace{.1in}}{\hspace{.1in}}{\hspace{.1in}} | \hspace{.5in} \}\text{.}\) This set may be empty, contain one ordered triple or more than one ordered triple.

Definition 1.5.2.

The three ways that can be used to transform a system of equations into a new system with exactly the same solutions are called elementary row operations. They are:

  1. rearrange the order of the rows

  2. multiply a row by a non-zero number

  3. add a multiple of one row to another

When working with systems of equations it is more efficient to write only the constants, making sure to keep them in columns so their variables can be reattached once the matrix is fully simplified. The result is called an augmented matrix.

\begin{equation*} \begin{array}{lcr} \left[ \begin{array}{ccccccc} a_{11}x \amp + \amp a_{12}y \amp + \amp a_{13}z \amp = \amp b_{1} \\ a_{21}x \amp + \amp a_{22}y \amp + \amp a_{23}z \amp = \amp b_{2} \\ a_{31}x \amp + \amp a_{32}y \amp + \amp a_{33}z \amp = \amp b_{3} \end{array} \right] \amp or \amp \left[ \begin{array}{ccccc} a_{11} \amp a_{12} \amp a_{13} \amp : \amp b_{1} \\ a_{21} \amp a_{22} \amp a_{23} \amp : \amp b_{2} \\ a_{31} \amp a_{32} \amp a_{33} \amp : \amp b_{3} \end{array} \right] \end{array} \end{equation*}

The elementary row operations apply to either the system of equations or the associated matrix. Ideally, but not always, the most simplified version looks like:

\begin{equation*} \begin{array}{lcr} \left[ \begin{array}{ccccccc} x \amp \amp \amp \amp \amp = \amp s_{1} \\ \amp \amp y \amp \amp \amp = \amp s_{2} \\ \amp \amp \amp \amp z \amp = \amp s_{3} \end{array} \right] \amp or \amp \left[ \begin{array}{ccccc} 1 \amp 0 \amp 0 \amp : \amp s_{1} \\ 0 \amp 1 \amp 0 \amp : \amp s_{2} \\ 0 \amp 0 \amp 1 \amp : \amp s_{3} \end{array} \right] \end{array} \end{equation*}

The solution in this case is a set with one ordered triple:

\begin{equation*} \mathbb{S} = \{ \ot{x}{y}{z}|\ x =s_1,y=s_2,z=s_3 \} = \{ \ot{s_1}{s_2}{s_3} \} \end{equation*}

Solve the following system of equations by transforming it into simpler and simpler systems of linear equations whose solution is identical to the given system.

\begin{equation*} \begin{array}{lcr} \left[ \begin{array}{ccccccc} 3x \amp + \amp 2y \amp - \amp z \amp = \amp 1 \\ x \amp + \amp y \amp \amp \amp = \amp 6 \\ \amp \amp -y \amp + \amp z \amp = \amp 0 \end{array} \right] \amp or \amp \left[ \begin{array}{ccccc} 3 \amp 2 \amp -1 \amp : \amp 1 \\ 1 \amp 1 \amp 0 \amp : \amp 6 \\ 0 \amp -1 \amp 1 \amp : \amp 0 \end{array} \right] \end{array} \end{equation*}

Solve the following system of equations. How many solutions are there?

\begin{equation*} \left[ \begin{array}{ccccccc} 2x \amp + \amp 4y \amp + \amp 6z \amp = \amp 8 \\ x \amp + \amp 3y \amp + \amp 5z \amp = \amp 7 \\ 2x \amp + \amp 5y \amp + \amp 8z \amp = \amp 10 \end{array} \right] \end{equation*}

Solve the following system of equations. How many solutions are there?

\begin{equation*} \left[ \begin{array}{ccccccc} 2x \amp + \amp 4y \amp + \amp 6z \amp = \amp 8 \\ x \amp + \amp 3y \amp + \amp 5z \amp = \amp 7 \\ 2x \amp + \amp 5y \amp + \amp 8z \amp = \amp 11 \end{array} \right] \end{equation*}

Write the solution in each of the following ways:

\begin{equation*} \mathbb{S} = \{\ \ot{x}{y}{z} \ | \; x=\hspace{.25in} ,\ y=\hspace{.25in} ,\ z=\hspace{.25in} \} \end{equation*}
\begin{equation*} \mathbb{S} = \{\ \ot{\hspace{.5in}}{\hspace{.5in}}{\hspace{.5in}}\ |\ t \in \mathbb{R}\} \end{equation*}
\begin{equation*} \mathbb{S} = \{\ \ot{\hspace{.25in}}{\hspace{.25in}}{\hspace{.25in}}+t\ot{\hspace{.25in}}{\hspace{.25in}}{\hspace{.25in}} \ | \ t \in \mathbb{R} \} \end{equation*}

Why are only the elementary row operations allowed when solving a system of equations? As a system of equations is simplified: What produces the possibility for no solutions? What produces the possibility for an infinite number of solutions? What might a fully reduced matrix look like in each case? Consider systems of two, three or four equations.