UNIT 6.15

Special Solutions: Graphing

I can identify special solutions within a system of equations graphically.

Vocabulary (review)

One Solution of a System of Linear Equations: The intersection of all lines in the system. Or the values of the variables that are true for all equations in the system.

Example:

(1, 2)

The solution of this linear system is (1, 2) or when x = 1, y =

2.

Vocabulary

No Solution: A System of Linear Equations can have no solution when there is no point of intersection of the lines (i.e. when the lines are… ).

Example:

There is no solution in this linear system

because there is no point of intersection!

Parallel

Vocabulary

Infinite Solutions: A System of Linear Equations can have infinite solutions when the equations describe the same line (i.e. when the lines … )

Example:

There are infinite solutions in this linear

system because the intersection includes ALL

points on the line!

Coincide

We already know we can solve Systems of Linear Equations

graphically. Let’s try it with a method I call the Graph-and-

check method.

First, you …

GRAPH IT …

(3, -1)

Then you CHECK IT!

(3, -1)When we substitute

the values of the variables back into

each equation, does it hold true?

Does -1 = -(3) + 2 ?

Does -1 = 3 – 4 ?

YesYes

GRAPH IT …

Infinite Solutions

!

Now, CHECK IT!

When we substitute in a point (x, y) that

works for one equation, does it

work for the other one?

Yes

Yes

Let’s try (1, -2).

Does (-2) + 1 = -(1) ?

Does (1) + (-2) = -1 ?

GRAPH IT …

No solution!

Now, CHECK IT!

Are the lines PARALLEL?

No

solution!

Graph & Check #1

(3, 5)

ClearBoards!

Graph & Check #2

NoSolution!

ClearBoards!

Graph & Check #3

No solution!

ClearBoards!

Graph & Check #4

(1, 3)

ClearBoards!

Graph & Check #5

Infinite solutions

!

ClearBoards!

6.15 GRAPHING SPECIAL SOLUTIONS WS

Homework Time!

I can identify special solutions within a system of equations graphically.