This document discusses systems of linear equations with two variables, including methods for solving them: substitution, elimination, graphing, and matrices. It identifies three possible outcomes for these systems: a single solution, no solution (inconsistent), or infinitely many solutions (dependent). Examples illustrate each method and outcome, complemented by classwork exercises.

1. 7.1 Systems of Linear Equations:
Two Variables
Chapter 7 Systems of Equations and Inequalities

2. Concepts and Objectives
⚫ Objectives for this section are
⚫ Solve systems of equations by graphing.
⚫ Solve systems of equations by substitution.
⚫ Solve systems of equations by addition.
⚫ Identify inconsistent systems of equations containing
two variables.
⚫ Express the solution of a system of dependent
equations containing two variables.

3. Systems of Linear Equations
⚫ A set of equations is called a system of equations. If all of
the variables in all of the equations are of degree one,
then the system is a linear system. In a linear system,
there are three possibilities:
⚫ There is a single solution that satifies all the
equations.
⚫ There is no single solution that satisfies all the
equations.
⚫ There are infinitely many solutions to the equations.

4. Systems of Linear Equations
⚫ There are four different methods of solving a linear
system of equations:
1. Substitution – Solve one equation for one variable,
and substitute it into the other equation(s).
2. Elimination – Transform the equations such that if
you add them together, one of the variables is
eliminated. Then solve by substitution.
3. Graphing – Graph the equations, and the solution is
their intersection.
4. Matrices – Convert the system into one or two
matrices and solve.

5. Substitution
Example: Solve the system.
1. Solving for y in the second equation will be simplest.
2. Now replace y with x + 3 in the first equation and solve
for x.
3 2 11
3
x y
x y
+ =
− + =
3
y x
= +
( )
3 2 11
3 2 6 11
5 5
1
3
x
x
x x
x
x
+ =
+ + =
=
=
+

6. Substitution (cont.)
3. Now we replace x in the first equation with 1 and solve
for y.
4. The solution is the ordered pair (1, 4). It is usually a
good idea to check your answer against the original
equations.
( )
4
1 3
y
y
− + =
=
( ) ( )
( )
3 1 2 4 11
1 4 3
+ =
− + =

7. Elimination (Addition)
Example: Solve the system.
While either variable would be simple to eliminate,
multiplying the second equation by 4 would be simplest.
4 3 13
5
x y
x y
+ = −
− + =
4 3 13
4 4 20
x y
x y
+ = −
− + =
7 7
1
y
y
=
=
Be sure to multiply
both sides!
1 5
4
4
x
x
x
− + =
− =
= − ( )
4,1
−

8. Inconsistent Systems
⚫ If after solving for the variables you end up with a false
statement (for example, 0 = 9), this is an inconsistent
system in that it has no solutions.
Example: Solve the system.
3 2 4
6 4 7
x y
x y
− =
− + =
6 4 8
6 4 7
0 15
x y
x y
− =
− + =
=

9. Infinitely Many Solutions
⚫ If after solving the variables you end up with a true
statement (ex. 0 = 0), then you have a dependent
system with infinitely many solutions. In this case,
you would express your solution set in terms of one of
the variables, usually y.
Example: Solve the system.
4 2
4 2
0 0
x y
x y
− = −
− + =
=
4 2
2
4
x y
y
x
= −
−
=
2
,
4
y
y
−
 
 
 

10. Classwork
⚫ College Algebra 2e
⚫ 7.1: 8-40 (x4); 6.7: 24-36 (even); 6.6: 68-76 (even)
⚫ 7.1 Classwork Check
⚫ Quiz 6.7