Solving Linear Systems Using the Substitution Method

Solving Linear Systems Using
the Substitution Method
Definition A linear system of equations is a set of two (or more)
linear equations containing the same variables.
A solution to a linear system is a point (a, b) that satisfies
all the equations in the system.
Main Idea Solving linear systems by substitution involves isolating one of
the variables in a single equation and then using that expression
to eliminate one of the variables in the other equation. The
exact procedure is below.
Procedure To solve a system by substitution:
1 Isolate one of the variables in one of the equations using reverse
operations.
→ Whenever possible, chose to isolate a variable that has a 1 or
negative 1 as its coefficient.
2 Substitute this expression into the other equation to eliminate a
variable.
3 Solve the resulting one variable equation.
4 Plug this number into any equation and solve for the remaining
variable.
Remark Solving systems by substitution is a reliable method, unlike the graphing
method which requires a certain amount of guessing (when determining
the intersection point). In most cases, the substitution method
is preferred over the graphing method.

Example 1 Solve the linear system by substitution. 3x + 5y = 25
x − 2y = −10
Example 2 Solve the linear system by substitution 1 1
x − y = 4
6 3
1 1
x + y = 0
4 2