We are multiplying the equations by 2 and 3 because we need to find the lowest common multiple for equation 1 and 2.
After multiplying both sides of the equation (we have to multiply both sides so the equation is still correct- remember to do so, its a common mistake), one of the equations have 24y and the other has -24 y.
We are able to add the two equations together to cancel out the y. After, we can solve them like a normal one variable equation.
For equations with more than 1 variable, need multiple equations. 2 variable, need to 2 equations, 3 needs 3, it goes on. Using multiple equations, we can add and subtract them, or substitute until we have a one variable equation where we can easily solve. Then we plug the variable back into the original equations to get the other variables.