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# Substitution Method

Substitution Method is an algebraic method to find the solution to a systems of equations. The is to replace, or substitute, a variable in any of the equations with an expression containing only the other variable. For example, the system of equations $\begin{cases}y-4=2x \\ 9x+6=3y \end{cases}$ be solved with the substitution approach in the following way.

Solving for one of the variables out of any equation so that it stands alone on one side of the equal sign. Then the expression on the other hand only contain the second variable. By adding 4 to both led in the first equation you can solve for $y$: $\begin{cases}y=2x+4 \\ 9x+6=3y. \end{cases}$

Replace the variable in the other equation with the expression that you got in the first step. The expression $2x+4$ is inserted instead of $y$ in the second equation: $\begin{cases}y=2x+4 \\ 9x+6=3({\color{#0000FF}{2x+4}}). \end{cases}$

Now, the second equation only one variable and it is possible to solve it with balance method.

$\begin{cases}y=2x+4 & \, \text {(I)}\\ 9x+6=3(2x+4) & \text {(II)}\end{cases}$
$\begin{cases}y=2x+4 \\ 9x+6=3\cdot2x+3\cdot4 \end{cases}$
$\begin{cases}y=2x+4 \\ 9x+6=6x+12 \end{cases}$
$\begin{cases}y=2x+4 \\ 3x+6=12 \end{cases}$
$\begin{cases}y=2x+4 \\ 3x=6 \end{cases}$
$\begin{cases}y=2x+4 \\ x=2 \end{cases}$

Insert the value of the variable for which you solved in the last step in any of the original equations. Usually, insert it in the expression to be solved out in the first step. To get the full solution to the system of equations to solve the last variable from the equation.

$\begin{cases}y=2x+4 & \, \text {(I)}\\ x=2 & \text {(II)}\end{cases}$
$\begin{cases}y=2 \cdot {\color{#0000FF}{2}}+4 \\ x=2 \end{cases}$
$\begin{cases}y=4+4 \\ x=2 \end{cases}$
$\begin{cases}y=8 \\ x=2 \end{cases}$

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