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Why does the elimination/addition method work?

When you solve a system of equations using the elimination/addition method, you add the sides of two equations with each other. In other words, the left side is equal to the sum of the right side.

Why can you do that? In equations you can add and subtract whatever you like as long as you do so on both sides. In the first equation, you can add $\text{VL}_2$ to both sides.

$\begin{cases}\text{VL}_1+{\color{#0000FF}{\text{VL}_2}}=\text{HL}_1+{\color{#0000FF}{\text{VL}_2}} \\ \phantom{\text{VL}_2 +}\text{VL}_2=\text{HL}_2 \end{cases}$

However, according the the other equation $\text{VL}_2$ is equal to $\text{HL}_2.$ You can substitute this into the right hand side of the first equation.

$\begin{cases}\text{VL}_1+\text{VL}_2=\text{HL}_1+{\color{#0000FF}{\text{HL}_2}} \\ \phantom{\text{VL}_2 +}\text{VL}_2=\text{HL}_2 \end{cases}$

The first equation is now exactly what you get when you add the equations column by column. In other words, the left hand side is equal to the sum of the right hand side. This is exactly what you do when you use the elimination/addition method!