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      <text>7.3.24 (Systems of equations with matrices)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h1>Solving simultaneous equations with matrices</h1>

<p>Now that we have seen how to perform basic operations with matrices let us return to our motivating problem of this section. Namely solving a system of three linear equations in three variables. Recall that we have the equations
\[
\begin{align*}
&a_1x_1+a_2x_2 + a_3x_3 = y_1
\\
&a_4x_1+a_5x_2 +a_6x_3 = y_2
\\
&a_7x_1+a_8x_2 +a_9x_3 = y_3,
\end{align*}.
\]
We can write these equations in terms of matrices as 
\[
\begin{bmatrix}
a_1 & a_2 & a_3
\\
a_4 & a_5 & a_6
\\
a_7 & a_8 & a_9
\end{bmatrix}
\begin{bmatrix}
x_1
\\
x_2
\\
x_3
\end{bmatrix} = \begin{bmatrix}
y_1
\\
y_2
\\
y_3
\end{bmatrix},
\]
Or, writing this in terms of matrices we have \(A\mathbf{x} = \mathbf{b}\)
This helps us solve these equations because we have translated the problem into a problem of matrices and we know that we have several techniques to deal with matrices. Recall that solving these equations means to find the value of each \(x_i\) variable. Notice that if we could find the inverse of the matrix \(A\) then we could multiply the equation on the left by \(A^{-1}\) to obtain
\[
A^{-1}A \mathbf{x} = A^{-1} \mathbf{b} \implies x = A^{-1} \mathbf{b},
\]
in other words we could solve the equation. Let's see this in the following example. 
</p>]]></text>
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