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

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.

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.

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

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
+ =
+ + =
=
=
+

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
+ =
? + =

Elimination (Addition)
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
?

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.
3 2 4
6 4 7
x y
x y
? =
? + =
6 4 8
6 4 7
0 15
x y
x y
? =
? + =
=

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.
4 2
4 2
0 0
x y
x y
? = ?
? + =
=
4 2
2
4
x y
y
x
= ?
?
=
2
,
4
y
y
?
? ?
? ?
? ?

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

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7.1 Systems of Linear Equations - Two Variables

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7.1 Systems of Linear Equations - Two Variables