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Algebra I
Chapter 6 Notes
Section 6-1: Graphing Systems of
linear equations, Day 1
What is a system of linear equations?
Consistent –
Inconsistent –
Independent –
Dependent –
Section 6-1: Graphing Systems of
linear equations, Day 1
What is a system of linear equations?
Two equations with two variables.
Consistent – a system with at least one solution
Inconsistent – a system with no solutions
Independent – a system with exactly one solution
Dependent – a system with an infinite number of
solutions
Section 6-1: Graphing Systems, Day 1
Number of
Solutions
Terminology
Graph
Exactly One
Infinite
No Solution
Section 6-1: Graphing Systems, Day 1
Steps for solving systems by graphing
Graph the system (Use a Ruler!)
y = -3x + 10
y=x–2
1. Graph the first equation
on the graph
2. Graph the second equation
on the graph
3. Find where the lines
intersect
4. CHECK your answer
Section 6-1: Graphing Systems, Day 1
Solve the following systems by graphing
Ex) y = ½ x
Ex) 8x – 4y = 16
y=x+2
-5x – 5y = 5
Section 6-1 Graphing Systems, Day 2
Systems that have no solutions –
Systems that have an infinite number of
solutions -
Section 6-1 Graphing Systems, Day 2
Systems that have no solutions –
Lines that are parallel and therefore never
intersect
Systems that have an infinite number of
solutions –
Equations that end up graphing the same line
Section 6-1 Graphing Systems, Day 2
Solve the following systems by graphing
Ex) 2x – y = -1
Ex) y = -2x - 3
4x – 2y = 6
6x + 3y = -9
Section 6-1 Graphing Systems, Day 2
Use the graph to determine whether each
system is consistent or inconsistent, independent
or dependent.
Ex) y = -2x + 3
y=x–5
Ex) y = -2x – 5
y = -2x + 3
Section 6-2: Solving Systems by
Substitution, Day 1
Steps for solving using
substitution:
1) Solve ONE equation for ONE
variable (Choose the a variable
with a coefficient of 1 or -1 to
make it easy)
2) Substitute the expression
from step 1 into the OTHER
equation for the variable
3) Solve the new equation
4) Plug in the solution from
step 3 into either equation to
find the other variable
5) Check your answer!
Ex) y = 2x + 1
3x + y = -9
Section 6-2: Solving Systems by
Substitution, Day 1
Solve the systems using substitution
Ex) y = x + 5
Ex) x + 2y = 6
3x + y = 25
3x – 4y = 28
Section 6-2: Solving Systems by
Substitution, Day 2
Special Case Solutions
Solve the systems using substitution
Ex) y = 2x – 4
Ex) 2x – y = 8
-6x + 3y = -12
-2x + y = -3
Section 6-3: Solving systems using the
elimination method (add/sub)
Steps for solving using the
elimination method
1) Write the system so like
terms are aligned
2) Add or subtract the
equations, elimination a
variable and solve
3) Plug in the solution from
step 2 to find the other
variable
4) Check your answer!
Ex) 4x + 6y = 32
3x – 6y = 3
Section 6-3: Solving systems using the
elimination method (add/sub)
Solve using elimination
Ex) 4y + 3x = 22
3x – 4y = 14
Ex) 7x + 3y = -6
7x – 2y = -31
Section 6-4: Elimination with
Multiplication, Day 1
Steps for solving using the
elimination method
1) Write the system so like terms
are aligned
2) Multiply one or both equations
by a number, or 2 different
numbers to get like coefficients
for one variable
3) Add or subtract the equations,
elimination a variable and solve
4) Plug in the solution from step 2
to find the other variable
5) Check your answer!
Ex) 5x + 6y = -8
2x + 3y = -5
Section 6-4: Elimination with
Multiplication, Day 1
Solve using the elimination method
Ex) 4x + 2y = 8
Ex) 6x + 2y = 2
3x + 3y = 9
4x + 3y = 8
Section 6-4: Solve using elimination,
Day 2
Solve using elimination. Be careful of special cases.
Ex) 3x + y = 5
Ex) x + 2y = 6
6x = 10-2y
3x + 6y = 8
Section 6-4: Solve using elimination,
Day 2
Solve the following systems using elimination
Ex) 8x + 3y = 4
Ex) 12x – 3y = -3
Ex) 8x + 3y = -7
-7x + 5y = -34
6x + y = 1
7x + 2y = -3
Section 6-5: Which method is best?
Method
Graphing
Substitution
Elimination
When to use it…
Section 6-5: Best Method
Determine which method is best, then solve the system using
that method
Ex) 2x + 3y = -11
Ex) 3x + 4y = 11
-8x – 5y = 9
y = -2x - 1
Section 6-5: Word Problems
Ex) Jenny has $24 to spend on tickets at
the carnival. The small rides cost $2 per
ticket, and the large rides cost $3 per
ticket. She buys a total of 7 tickets. How
many small ride tickets did she buy?
How many large ride tickets did she
buy? Write and solve a system.
Ex) Martha has a total of 40 DVDs of
movies and TV shows. The number of
movies is 4 less than 3 times the
number of TV shows. Write and solve a
system to find the numbers of movies
and TV shows she owned.
Section 6-6: Systems of Linear
Inequalities
Steps for Solving Systems of
Linear Inequalities
1) Graph the first equation
• Choose the correct line
(Solid or dashed)
• Shade the correct side
2) Graph the second equation
• Choose the correct line
(Solid or dashed)
• Shade the correct side
3) Darken the shaded areas
that overlap
Ex) y > -2x + 1
y<x+3
Section 6-6: Systems of Linear
Inequalities
Solve the following S.o.L.E by graphing
Ex) x > 4
Ex) y > -2
y<x–3
y<x+9
Section 6-6: Graphing systems of linear
inequalities
Solve the following S.o.L.E by graphing
Ex) 3x – y > 2
Ex) y > 3
3x – y < -5
y<1