Download system of linear inequalities

Solving Systems of
Equations
Chapter 5
Solving Systems of Linear
Equations by Graphing
I can solve systems of linear equations by
graphing.
Solving Systems of Linear
Equations by Graphing
Vocabulary (page 134 in Student Journal)
system of linear equations: two or more
linear equations with related variables
solution of a system of linear equations:
any ordered pair that makes all of the equations
in the system true
Solving Systems of Linear
Equations by Graphing
Examples (space on page 134 in Student
Journal)
Determine if the ordered pair is a solution to
the system of equations.
a) (5, 1)
b) (-1, -5)
x+y=6
y = -x + 2
2x – y = 9
y = 3x - 2
Solving Systems of Linear
Equations by Graphing
Solutions
a) yes
b) no
Solving Systems of Linear
Equations by Graphing
Solve the system by graphing.
c)
Solving Systems of Linear
Equations by Graphing
Solution
c) (2, 3)
Solving Systems of Linear
Equations by Graphing
Example
d) Solve the system of equations by graphing.
y = -2x + 5
y = 4x - 1
Solving Systems of Linear
Equations by Graphing
Solution
d)
(1, 3)
Solving Systems of Linear
Equations by Graphing
e) A museum sells 4 children’s tickets and 8
adult tickets for their morning show and makes
$128. For the afternoon show, they make $72
by selling 6 adult tickets. What is the price of a
children’s ticket?
Solving Systems of Linear
Equations by Graphing
Solution
e) 4x + 8y = 128
6y = 72
x=8
$8 per child’s ticket
Solving Systems of Linear
Equations by Substitution
I can solve systems of linear equations by
substitution.
Solving Systems of Linear
Equations by Substitution
Core Concepts (page 139 in Student Journal)
Steps for using Substitution Method
1. solve one equation for a variable
2. substitute the expression into the other
equation for the variable and solve the new
equation for the variable
3. substitute value of variable into either
equation and solve for the other variable
Solving Systems of Linear
Equations by Substitution
Examples (space on page 139 in Student
Journal)
Solve the system by substitution.
a) y = -x + 3
3y + 5x = -1
b) x – y = -4
2x + y = 4
Solving Systems of Linear
Equations by Substitution
Solutions
a) (-5, 8)
b) (0, 4)
Solving Systems of Linear
Equations by Substitution
c) An theater earns $3640 from a concert. An
orchestra ticket costs 3 times as much as a
balcony ticket. Below are the sales from a
recent concert. What is the price of each type of
ticket?
Solving Systems of Linear
Equations by Substitution
Solution
c) y = 3x
148y + 76x = 3640
x = 21, y = 7
$21 orchestra ticket, $7 balcony ticket
Solving Systems of Linear
Equations by Elimination
I can solve systems of linear equations by
elimination.
Solving Systems of Linear
Equations by Elimination
Core Concepts (page 144 in Student Journal)
Steps for using Elimination Method
1. if necessary, multiply one or both equations
by a constant
2. add or subtract to eliminate a variable
3. solve the resulting equations
4. substitute value of variable into either
equation and solve for the other variable
Solving Systems of Linear
Equations by Elimination
Examples (space on page 144 in Student
Journal)
Solve the system by elimination.
a) -2x + 3y = 4
2x – y = -8
b) 3x + 2y = -2
-6x – 5y = 2
Solving Systems of Linear
Equations by Elimination
Solutions
a) (-5, -2)
b) (-2, 2)
Solving Systems of Linear
Equations by Elimination
c) A shipping business has 2 locations. Location
A has 3 large trucks and 2 small trucks, which
cost $270,000. Location B has 4 large trucks
and 3 small trucks, which cost $375,000. What
is the cost of each type of truck?
Solving Systems of Linear
Equations by Elimination
Solution
c) 3x + 2y = 270,000
4x + 3y = 375,000
(60000, 45000)
$60,000 for the large truck and $45,000 for
the small truck
Solving Special Systems of
Linear Equations
I can determine the numbers of solutions of
linear systems.
Solving Special Systems of
Linear Equations
Core Concepts (page 149 in Student Journal)
We will get 1 solution when the lines intersect.
We will get no solutions when the lines are
parallel.
We get infinitely many solutions when the lines
are the same.
Solving Special Systems of
Linear Equations
Examples (space on page 149 in Student
Journal)
Solve the system of linear equations.
a) y = 3x + 2
y = 3x – 1
b) x – 3y = 6
3x – 9y = 18
Solving Special Systems of
Linear Equations
Solutions
a) no solution
b) infinitely many solutions
Graphing Linear Inequalities
in Two Variables
I can graph linear inequalities in 2 variables.
Graphing Linear Inequalities
in Two Variables
Vocabulary (page 159 in Student Journal)
linear inequality in two variables: formed
by replacing the equal sign in a linear equation
with an inequality symbol
solution of a linear inequality: an ordered
pair that makes the inequality true
Graphing Linear Inequalities
in Two Variables
graph of a linear inequality: shows all of
the solutions of the inequality in the coordinate
plane
half-plane: a region that is bounded by a line.
Graphing Linear Inequalities
in Two Variables
Core Concepts (page 159 in Student Journal)
Guidelines for Graphing Linear Inequalities
greater than/less than: use dashed line
greater than or equal to/less than or equal to:
use solid line
greater than/greater than or equal to: shade
above line
less than/less than or equal to: shade below line
Graphing Linear Inequalities
in Two Variables
Examples (space on page 159 in Student
Journal)
Determine whether the ordered pair is a
solution to the inequality.
a) 3x – y < 2; (-2, 2)
b) 4x – y > 5; (1, 3)
Graphing Linear Inequalities
in Two Variables
Solutions
a) yes
b) no
Graphing Linear Inequalities
in Two Variables
Graph in the coordinate plane.
c) y ≥ -3
d) x + y < 2
Graphing Linear Inequalities
in Two Variables
Solutions
c)
d)
Graphing Linear Inequalities
in Two Variables
e) You can spend at most $9 for potatoes and
carrots for a stew you are making. Potatoes are
$3 a pound and carrots are $1.50 per pound.
Write and graph an inequality that represents
the amount of potatoes and carrots you can buy
for the stew.
Graphing Linear Inequalities
in Two Variables
Solution
e) 3x + 1.5y ≤ 9
Systems of Linear
Inequalities
I can graph systems of linear inequalities.
Systems of Linear
Inequalities
Vocabulary (page 164 in Student Journal)
system of linear inequalities: made up of 2
or more linear inequalities
solution to a system of linear
inequalities: an ordered pair that makes all
the inequalities in the system true
Systems of Linear
Inequalities
Examples (space on page 164 in Student
Journal)
Determine whether each ordered pair is a
solution to the system of inequalities.
a) y > 3x
b) y > 3x
y ≤ -x – 2
y ≤ -x – 2
(-2, -1)
(0, 4)
Systems of Linear
Inequalities
Solutions
a) yes
b) no
Systems of Linear
Inequalities
Graph the system of inequalities.
c) y ≥ -1
y < -x + 2
d) x – y > 2
x–y≤-3
Systems of Linear
Inequalities
Solutions
c)
d)
no solution
Systems of Linear
Inequalities
Write the system of inequalities from the graph.
e)
f)
Systems of Linear
Inequalities
Solutions
e) y ≤ 2
y > -x + 1
f) x ≥ 1
y ≥1/2 x - 2
Systems of Linear
Inequalities
g) You have at most 7 hours to spend swimming
and playing soccer. You want to spend at least 2
hours playing soccer and you want to spend
more than 2 hours swimming. Write and graph
a system to determine how much time you can
spend on each activity.
Systems of Linear
Inequalities
Solution
g) x + y ≤ 7
x≥2
y>2