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Substitution method

The substitution method is an algebraic method for finding a solution to a system of equations. The method replaces or substitutes a variable in one of the equations with an expression that contains the other variables. For example, the equation system {y4=2x9x+6=3y\begin{cases}y-4=2x \\ 9x+6=3y \end{cases} can be solved using the substitution method by doing the following:

Solve for one of the variables in one of the equations so that it stands alone on one side of the equals sign. This way the expression on one side only includes the second variable. By adding 4 to each side in the first equation, you can solve for yy: {y=2x+49x+6=3y.\begin{cases}y=2x+4 \\ 9x+6=3y. \end{cases}

Replace the variabel in the second equation with the expression you received from the step one. The expression 2x+42x+4 is used instead of yy in the other equation. {y=2x+49x+6=3(2x+4).\begin{cases}y=2x+4 \\ 9x+6=3({\color{#0000FF}{2x+4}}). \end{cases}

The second equation now contains only one variable and can be solved by balancing out the equation.

{y=2x+4(I)9x+6=3(2x+4)(II)\begin{cases}y=2x+4 & \, \text {(I)}\\ 9x+6=3(2x+4) & \text {(II)}\end{cases}
{y=2x+49x+6=32x+34\begin{cases}y=2x+4 \\ 9x+6=3\cdot2x+3\cdot4 \end{cases}
{y=2x+49x+6=6x+12\begin{cases}y=2x+4 \\ 9x+6=6x+12 \end{cases}
{y=2x+43x+6=12\begin{cases}y=2x+4 \\ 3x+6=12 \end{cases}
{y=2x+43x=6\begin{cases}y=2x+4 \\ 3x=6 \end{cases}
{y=2x+4x=2\begin{cases}y=2x+4 \\ x=2 \end{cases}

Substitute the value calculated in step one into either of the original equations. Often you put the value into the expression you solved for in step one. In order to get the complete solution for the equation system, you then need to calculate the other variable using the equation.

{y=2x+4(I)x=2(II)\begin{cases}y=2x+4 & \, \text {(I)}\\ x=2 & \text {(II)}\end{cases}
{y=22+4x=2\begin{cases}y=2 \cdot {\color{#0000FF}{2}}+4 \\ x=2 \end{cases}
{y=4+4x=2\begin{cases}y=4+4 \\ 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}