Substitution
Substitution is when one replaces all instances of a variable (or expression) with another equivalent variable (or expression).
Contents
Uses
System of Equations
- Main article: System of equations
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.
For example, consider the below system.
![\[\left\{\begin{array}{l}x+y=-1\\3x-y=5\end{array}\right.\]](http://latex.artofproblemsolving.com/1/1/7/117a178c31d38a5d1b1c201fc274295f64f9fc66.png)
An example of solving the system by substitution is when we start by isolating 
in the top equation to get 
. Then, we can replace all instances of with 
in the second equation. Doing so results in an equation with one variable, and solving it results in
After solving for 
, we can "plug in" the value 
for to get 
, so the solution to the system is 
. As usual, we can check by substituting for and 
for .
This same method is used for simultaneous equations with more than two equations.
Observing Common Parts
Substitution can also be used when an expression has multiple common parts.
For instance, consider the equation 
. Note that with exponent properties, we can rewrite the equation as 
. Because there are multiple instances of 
in the equation, we can let 
and substitute for to make the equation easier to solve.
Doing so results in
Finally, we can substitute for to get 
or 
, resulting in 
or 
. As usual, we can check by substituting and 
for separately.
