6-1.1 – Identifying Solutions of Systems
Vocabulary:
System of linear equations – a system of equations in which all of the equations are linear
Solution of a system of linear equation – any ordered pair that satisfies all the equations in a system
A system of linear equations are 2 or more linear equation that are related to each other in some way.
A lot of times we want to be able to solve these systems simultaneously or at the same time. When that happens we need to find a value for x and y that will make both equation true at the same time.
Examples: Tell whether the ordered pair is a solution of the given system
 
; x

2 y x
 y


3
6 

1 , 2

;
2 x
3 x


5 y
2 y

8

5
Plug the x and y values into both equation and see if they work.
( 4 )

2 ( 1 )

6
4

2

6
4

1

3
3

3
6

6
Both are true so it is a solution
2 (

1 )

5 ( 2 )

8

2

10

8
3

(

3
1

)
4


2 (
5
2 )

5
8

8

7

5
Both are not true so it is not a solution
6-1.2 – Solving a system of linear equations by graphing
All of the ordered pairs that make a linear equation true are on its graph. So we need to find a point that is on both of the line graphs, in other words we need the point of intersection of the two lines.
Examples: Solve each system by graphing y
 y
 x

 x
3

1
The lines cross at
 
so that is the solution x
 y

0 y
 
1
2 x

1
The lines cross at


2 , 2

so that is the solution

6-1.3 – Problem solving application
Bowl-o-Rama charges $2.50 per game plus $2 for shoe rental, and Bowling Plaza charges $2 per game plus $4 for shoe rental. For how many games will the cost to bowl be the same at both places?
Write the equations: Bowl-a-Rama:
Bowling Plaza: y y

2 .
50 x

2 x


2
4
Graph both linear equations:
The numbers of games would be 4 for a cost of $12
6-1.4 – Systems with no solutions or infinite solutions
Vocabulary:
Consistent System – Systems of Equations or Inequalities with at least one solution
Inconsistent System – A system of equations or inequalities with no solutions
Independent System – A system of equations that has exactly one solution
Dependent System – A system of equation that has infinitely many solutions
We graphed the 2 lines in a system and saw they intersected. This intersection was the solution to the system.
However, what if the lines were parallel? Then they would not intersect and there would not be a solution. We call these systems Inconsistent Systems.
If the two equations in a system have the same graph (meaning that one lies on top of the other) then there are infinitely many solutions because they share all the same points. These are said to be Dependent Systems. All
Consistent Systems are either Dependent or Independent.
Examples: 𝑦 = 2𝑥 + 2 𝑦 = 2𝑥 − 1
2𝑦 − 𝑥 = 2 𝑦 =
1
2 𝑥 + 1 𝑦 = 𝑥 + 2
1 𝑦 = −
2 𝑥 +
1
2
Inconsistent Consistent and Dependent Consistent and Independent