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Elimination/Addition Method

The elimination/addition method is an algebraic method för finding solutions for a system of equations. This method words by "eliminating" or cancelling out one variable by adding the equations column by column. For example the system of equations can

{y4=2x9x+6=3y\begin{cases}y-4=2x \\ 9x+6=3y \end{cases}

be solved using the elimination method by doing the the following:

By rearranging terms in the equations to have the same order, you can more easily compare two equations. In the example, the terms containing variables are moved the the left hand side and the terms with only constants to the right hand side.

{y4=2x(I)9x+6=3y(II)\begin{cases}y-4=2x & \, \text {(I)}\\ 9x+6=3y & \text {(II)}\end{cases}
{-4=2xy9x+6=3y\begin{cases}\text{-}4=2x-y \\ 9x+6=3y \end{cases}
{2xy=-49x+6=3y\begin{cases}2x-y=\text{-}4 \\ 9x+6=3y \end{cases}
{2xy=-49x=3y6\begin{cases}2x-y=\text{-}4 \\ 9x=3y-6 \end{cases}
{2xy=-49x3y=-6\begin{cases}2x-y=\text{-}4 \\ 9x-3y=\text{-}6 \end{cases}

Now, you want the coefficients in front of one of the variables to be the same in both equations, but with the opposite sign. You do this by multiplying both sides of an equation with an appropriate constant. In the example, equation (I) is multiplied by -3\text{-} 3 in order to get the term 3y3y in equation (I) and -3y\text{-} 3y in equation (II). {-6x+3y=129x3y=-6\begin{cases}\text{-} 6x + 3y=12 \\ 9x-3y=\text{-}6 \end{cases}

The equations are then added column by column. This means that the left side of an equation is added to the left side of the other and the right hand side of one equation is added to the other.

Two equations which are added column by column

Solve the equations by balancing them in order to determine one of the variables.

{9x3y=-6(I)3x=6(II)\begin{cases}9x - 3y=\text{-} 6 & \, \text {(I)}\\ 3 x=6 & \text {(II)}\end{cases}
{9x3y=-6x=2\begin{cases}9x - 3y=\text{-} 6 \\ x=2 \end{cases}

You can then substitute the value of the now known variable into either of the original equations, determine the other variable, and get the entire solution for the system of equations. In the example, x=2x=2 is inserted into equation (I).

{9x3y=-6(I)x=2(II)\begin{cases}9x - 3y=\text{-} 6 & \, \text {(I)}\\ x=2 & \text {(II)}\end{cases}
{923y=-6x=2\begin{cases}9\cdot {\color{#0000FF}{2}} - 3y=\text{-} 6 \\ x=2 \end{cases}
{183y=-6x=2\begin{cases}18 - 3y=\text{-} 6 \\ x=2 \end{cases}
{-3y=-24x=2\begin{cases}\text{-} 3y=\text{-}24 \\ x=2 \end{cases}
{y=8x=2\begin{cases}y=8 \\ x=2 \end{cases}

The solution for the system of equations is {x=2y=8.\begin{cases}x=2 \\ y=8. \end{cases}