7.5 Special Types
of Linear Systems
The equations you have
been waiting for have
finally arrived!
Focus 5 Learning Goal – (HS.A-CED.A.3, HS.A-REI.C.5, HS.A-REI.C.6, HS.AREI.D.11, HS.A-REI.D.12): Students will write, solve and graph linear
systems of equations and inequalities.
4
3
2
1
0
In addition to
level 3.0 and
above and
beyond what was
taught in
class, the student
may:
· Make
connection with
other concepts in
math
· Make
connection with
other content
areas.
The student will write,
solve and graph systems of
equations and inequalities.
- Solve systems of linear
equations graphically, with
substitution and with
elimination method.
- Solve systems that have
no solutions or many
solutions and understand
what those solutions mean.
- Find where linear and
quadratic functions
intersect.
- Use systems of equations
or inequalities to solve real
world problems.
The student will
be able to:
- Solve a
system
graphically.
- With help the
student will be
able to solve a
system
algebraically.
With help from
the
teacher, the
student has
partial success
with solving a
system of linear
equations and
inequalities.
Even with
help, the
student has no
success
understanding
the concept of
systems of
equations.
Special linear systems
Intersecting
One solution
(x, y)
Parallel
Same line
No solution Many solutions
0= 0
0=2
When you solve each system, you
either get an ordered pair, a false
statement, or both sides are equal.
Solve by substitution or combination then
graph to check.
3x – 2y = 3
-6x + 4y = -6
Multiply the top equations by 2
6x – 4y = 6
-6x + 4y = -6
0 = 0 (true)
What does this mean?????
Rewrite in slope-intercept form:
y = mx + b
3x – 2y = 3
-6x + 4y = -6
y = 3/2x -3/2
y = 3/2x -3/2
You have the same equations, so you
have the same line and infinite
solutions!
You can graph to check.
Infinite solutions
Same line
False Statement
Parallel lines
Solve by substitution or combination then graph.
3x – 2y = 12 Multiply top by 2
-6x + 4y = -12
6x - 4y = 24
-6x + 4y = -12
0 = 12 (False)
Rewrite in slope-intercept form:
3x – 2y = 12
-6x + 4y = -12
y = 3/2x -6
y = 3/2x -3
Notice, same slope but different yintercepts. You have parallel lines with
NO solution. They will never intersect!
Find a linear system for the graphical model.
If only one line is shown, find two
different equations for the line.
• a: y = 3/2x + 1
• b: y = 3/2x - 1
•
•
•
•
y = 2x - 4
6x – 3y = 12 or
12x – 6y = 24 or
18x – 9y = 36 or…
One More Time!
Special linear systems:
Intersecting
One solution
(x, y)
Parallel
Same line
No solution Many solutions
0= 0
0=2
When you solve each system, you
either get an ordered pair, a false
statement, or both sides are equal.