Example VESE Vector equality for a system of equations

Consider the system of linear equations in Archetype B, \begin{align*} -7x_1 -6 x_2 - 12x_3 &=-33\\ 5x_1 + 5x_2 + 7x_3 &=24\\ x_1 +4x_3 &=5 \end{align*}

Note the use of three equals signs — each indicates an equality of numbers (the linear expressions are numbers when we evaluate them with fixed values of the variable quantities). Now write the vector equality, \begin{equation*} \colvector{-7x_1 -6 x_2 - 12x_3\\ 5x_1 + 5x_2 + 7x_3\\ x_1 +4x_3} = \colvector{-33\\24\\5}. \end{equation*} By Definition CVE, this single equality (of two column vectors) translates into three simultaneous equalities of numbers that form the system of equations. So with this new notion of vector equality we can become less reliant on referring to systems of simultaneous equations. There is more to vector equality than just this, but this is a good example for starters and we will develop it further.