MTH 100
Systems of Linear Equations in Two
Variables
Objectives
1. Solve Systems of Linear Equations in Two
Variables by Substitution.
2. Solve Systems of Linear Equations in Two
Variables by Elimination.
3. Solve Inconsistent and Dependent Systems.
Overview
• A system of linear equations in two variables
looks like this:
 Ax  By  C

Dx

Ey

F

• Since linear equations make straight lines, the
lines produced from two equations have three
possibilities for their interaction:
1. The lines intersect in a single point.
• There is one ordered pair solution. We call this
type of system independent.
2. The lines are parallel.
• There are no ordered pair solutions. We call
this type of system inconsistent.
3. The lines are actually the same line.
• There are infinitely many ordered pair
solutions. We call this type of system
dependent.
Solving Methods
1. Graphing (we don’t do this one)
2. Substitution (one of the equations has an
isolated, or easily isolated, variable)
3. Elimination (add the equations together and
cancel one of the variables)
Examples
6 x  3 y  33

x  2 y  2
5 x  4 y  10

 4 x  3 y  23
 y  4 x

8 x  2 y  4
x  4 y  2

4 x  16 y  8