# Substitution

Substitution is a relatively universal method to solve simultaneous equations. It is generally introduced in a first year high school algebra class. A solution generally exists when the number of equations is exactly equal to the number of unknowns. The method of solving by substitution includes:

1. Isolation of a variable 2. Substitution of variable into another equation to reduce the number of variables by one 3. Repeat until there is a single equation in one variable, which can be solved by means of other methods.

Example:

Solve $x+y=-1, 3x-y=5$ for $(x,y)$.

Start with $x+y=-1$. $x+y=-1$ Subtract $x$ from both sides. $y=-x-1$ $y$ is now isolated.

Substitute $(-x-1)$ for the y in $3x-y=5.$ $3x-(-x-1)=5$ Distribute the negative sign. $3x+x+1=5$ Combine like terms. $4x+1=5$ Subtract 1 from both sides. $4x=4$ Divide both sides by four. $x=1$

x is now solved for, so substitute x into one of the original equations. $1+y=-1$ Subtract 1 from both sides. $y=-2$ $(x,y)=(1,-2)$

You can check this answer by plugging x and y into the original equations.

This same method is used for simultaneous equations with more than two equations.

Invalid username
Login to AoPS