Substitution

From AoPSWiki

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 Subtact 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.