I have a very simple linear problem:

$\begin{array}{rl}\underset{x}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}+{a}_{2}{x}_{2}=b\end{array}$

Suppose I want to write this problem equivalently as in Find the equivalent linear program. Unlike the problem in the link, I have equality. Can I write it equivalently as:

$\begin{array}{rl}\underset{x,\alpha ,\beta}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}=\alpha b,\text{}{a}_{2}{x}_{2}=\beta b,\text{}\alpha +\beta =1.\end{array}$

The converse is intuitive: Given $\{x,\alpha ,\beta \}$ feasible for the second problem, adding the first and second constraints gives the constraint of the first problem. But the forward part is not clear, especially because I have never seen an equality constraint written like this. Any help would be highly appreciated.

$\begin{array}{rl}\underset{x}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}+{a}_{2}{x}_{2}=b\end{array}$

Suppose I want to write this problem equivalently as in Find the equivalent linear program. Unlike the problem in the link, I have equality. Can I write it equivalently as:

$\begin{array}{rl}\underset{x,\alpha ,\beta}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}=\alpha b,\text{}{a}_{2}{x}_{2}=\beta b,\text{}\alpha +\beta =1.\end{array}$

The converse is intuitive: Given $\{x,\alpha ,\beta \}$ feasible for the second problem, adding the first and second constraints gives the constraint of the first problem. But the forward part is not clear, especially because I have never seen an equality constraint written like this. Any help would be highly appreciated.