Chapter 6: Systems of Linear Equations
Graphing Systems of Equations
Notes 6.1
Systems of Equations: __________________________________________________________________
*Solution: ____________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Three ways:
1.)
2.)
3.)
Solve by Graphing: _____________________________________________________________________
Ex: Solve the following SoE by graphing:
Ex: Solve the following SoE by graphing:
y = 3x + 1
3x + y = 4
and
y = -x + 5
and
x – 2y = 6
Solve the following SoE’s by graphing:
1.) y + 2x = 2
y+x=1
3.) y – 3x = 6
-9x = -3y + 18
2.) x + 7 = y
-4x + 2 = y
4.) y = -2x + 3
y = 8 – 2x
The Substitution Method
Notes 6.2
You can find an exact solution to a system of linear equations without graphing. One method of doing
this is to use the substitution method.
Substitution: __________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Solve the following SoE by substitution:
Ex: 8x + 2y = 9
x=3
Ex: 15x – 5y = 30
y = 2x + 3
Solve the following SoE’s by substitution:
1.) 2x + 5y = 14
y=5
3.) 3x + y = 4
5x – 7y = 11
2.) -3x + 2y = 31
x = 0.5y + 6
4.) 6x – 2y = 11
x + 3y = 4
The Elimination Method
Notes 6.3
What does it mean to eliminate something?
Elimination: This method uses opposites to eliminate one of the variables. You will basically be adding
the two equations that make up the system of equations, and trying to make one of the variables
eliminate themselves in the process.
Solve the following by elimination:
Steps:
3x + 4y = 7
1.) Look for an “x” or “y” that are
2x – 4y = 13
already opposites of each other.
* Sometimes you have to make opposites by multiplying one or both equations by a value to make the
coefficients of either the x or the y eliminate themselves when added together.
3x + 4y = 7
2.) Set up the addition problem.
+ 2x – 4y = 13
5x
= 20
5x = 20
3.) Solve for the remaining variable.
3(4) + 4y = 7
4.) Plug in your value into one of the
12 + 4y = 7
4y = -5
original equations to solve for the other.
x = 4 plugged into 3x + 4y = 7
y = -1.25
Solution:
(4, -1.25)
5.) Write your answer as an ordered pair!
Ex: Solve using Elimination:
2x + 3y = 1
5x + 7y = 3
Ex: Solve using Elimination:
2x – 5y = -20
4x + 5y = 14
* You will need to multiply the equations
by a value to eliminate one variable.
Ex: Solve using Elimination:
9a – 2b = -11
8a – 4b = 25
Think about the following situation:
Jason rented a car for two days traveling 125 miles and paying $95.75. He rented a different car for four
days traveling 350 miles and paying $226.50. To find the daily fee and the per mile cost at the Airport
Rent – a – Car, you can set up and solve a system of equations.
Define the variables, let d represent the dost per day, and let m represent the cost per mile.
Two day rental equation: ________________________________________
Four day rental equation: ________________________________________
Both the graphing and substitution methods provide options for finding the solution to this system.
However, neither of these methods would be easy to apply to this system. Elimination would be a much
better method to solve.
Solving Special Systems
Notes 6.4
Consistent: Systems that have _________________________________________.
Inconsistent: Systems that have _____________________________________.
Consistent Systems
One unique Solution (x, y)
Inconsistent Systems
Infinitely many solutions
No solution
x + 2y = 3
x+y=3
-2x + y = -4
2x – y = 1
3x + 3y = 9
-2x + y = 3
These lines intersect
These lines are the same
Independent
Dependent
These lines are parallel
Ex: Determine if the following system is consistent or inconsistent by solving it through both graphing
and substitution:
y=4–x
y = 3x – 6
Substitution:
Graphing:
Ex: Determine if the following system is consistent or inconsistent by solving it through both graphing
and elimination.
2x – y = -1
4x – 2y = 4
Elimination:
Graphing:
*Consistent systems of equations can be divided further into two categories, independent and
dependent.
Independent System: Has only one solution, only one unique ordered pair (x, y) that satisfies both
equations.
Ex: x + 2y = 3
2x – y = 1
Dependent System: Has infinitely many solutions, every ordered pair that is a solution of the first
equation is also a solution of the second equation.
Classify the following systems as inconsistent, consistent, independent or dependent based on
their “y = “equations. State why you labeled them as such.
1.) y = 5x – 3
2.) x + 2y = -4
y = 3x – 5
-2(y + 2) = x
3.) y = -2(x – 1)
4.) 2x – 3y = 6
2
y=(
y = -x + 3
)x
3
Create a system that satisfies the following:
5.) Consistent & Dependent
6.) Consistent & Independent
7.) Inconsistent
Solving Linear Inequalities
Notes 6.5
A linear inequality is like a linear equation, but the equal sign is replaced with an inequality symbol. A
solution of a linear inequality is any ordered pair that makes the inequality true. A few things need to be
kept in mind:
1.) When solving for the “y =” part, remember when your inequality sign may need to flip.
(__________________________________________________________________________________)
2.) Boundary line: Solid if it says: ____________________________________________
Dotted if it says: __________________________________________
3.) Pick a nice point to test (The nicest point to test is (0, 0) however you can’t use this point if it….
_________________________________________________________, shade where _______________!)
Graph the following:
1.) y < 3x + 4
2.) 4x – 3y > 12
3.) y > (−
2
)x + 1
3
Solving Systems of Linear Inequalities
Notes 6.6
The inequalities we studied in chapter 4 were graphed on a number line because they only had one
variable. In this chapter, we will study inequalities that have two variables, and graphed on a Cartesian
plane.
Linear Inequality: In the form of a linear equation where the equal sign is replaced with an inequality
sign. When graphed, the ordered pairs on the half that make up the solution are shaded.
Boundary Line: The line that divides the coordinate plane into two half planes. The boundary line is
either dotted or solid, based on the sign.
< , > : Solid line, the values could exist or be equal to those on the line.
< , > : Dotted line, the values could not exist or be equal to those on the line.
Ex: Graph and solve:
x – 2y < 4
-x
-x
Steps:
1.) Solve for y
* adding/subtracting doesn’t change sign
-2y < 4 – x
* multiplying/dividing by a pos. doesn’t change sign
-2
* multiplying/dividing by a neg. does change sign
-2
1
y > -2 + x
2
2.) Graph!
< , > : Solid boundary line
< , > : Dotted boundary line
x – 2y < 4, test point (0, 0)
3.) Pick a point in either region, and test for the
(0) – 2(0) < 4
solution! The point (0, 0) works well! Easy
0–0<4
math! If the statement is true, shade where
0<4
that point lies!
When solving a system of linear inequalities, it is much like solving systems of linear equations, but using
inequalities instead of equations. The solution of a system of linear inequalities is the intersection of
the solutions of each inequality. Every point in the intersection region satisfies the system. To test and
make sure, pick a point inside the overlapping shaded region and check that point in both inequality
statements.
Ex: Solve by Graphing:
1
y<−(
2
Steps:
)x – 1
1.) Graph the first inequality, test and shade
y>x+3
2.) Graph the second inequality, test and shade.
3.) Solution: the overlapping shaded regions!
1
y<−(
2
)x – 1
y>x+3
Solution!
Overlapping Shaded region = Solution
Solve the following by graphing:
1.) 4x + y < 7
6 > -2x – y
2.) y > x – 3
y<x+1
3.) 3x + y > 3
4x + 3y < 9
4.) y < 2x – 3
y > 2x + 2