3.1 Solving Systems by Graphing or Substitution
System of Equations
{
y=x-3
y = 5 - 3x
One way to solve a system of equations is
to graph both and the intersection point is
the values for x and y. The values for x
and y also make each equation true when
substituted in.
{
y=x-3
y = 5 - 3x
{
3x - y = -4
-4x - 2y = -8
{
3x - y = -4
-4x - 2y = -8
{
x+y=5
x - 5y = -7
1
{
x+y=5
Assignment
x - 5y = -7
3.1
3.1 Solving Systems Using Substitution
Warm-up mystery:
If
2
5
2
page 161
#13-24 graphs and solutions only!
-1
{
y = 4x -2
y=x+7
Then what does frog and armadillo
equal numerically?
4
4
2
Please solve this solution using substitution.
Solve this system. Solve it good.
{
2x + y = 3
{
3x + y = 8
3x - 2y = 8
18x + 2y = 4
Solve this system. Solve it good.
{
3x + y = 8
18x + 2y = 4
{
The Monster Problem
x+y+z=5
2x - 3y + z = -2
4z = 8
The Monster Problem
{
x+y+z=5
2x - 3y + z = -2
Assignment
3.1
pages 161-162
You need to use substitution.
No graphing!
4z = 8
#25-33, 41-42
3
3.2 Solving Systems by Elimination
Warm‐up:
Solve the system for x and y any way possible.
{
3.2 Using the elimination method to
solve systems of equations
1) Use multiplication or division to
transform the system of equations so
that the coefficients of one of the
variables are the same
2) Subtract the equations to eliminate
one of the variables and solve for the
remaining variable
3) Substitute this value back into
either of the equations to find y.
4) Check both answers in both
equations.
Solve using the elimination method. Or else...
{
2x + 5y = 15
-4x + 7y = -13
{
5.5x + 7.5y = 930
12x + 15y = 1920
5.5x + 7.5y = 930
12x + 15y = 1920
Eliminate y
Solve for y
Check:
Use elimination to solve the system. I'm sorry for threatening you. I didn't mean it.
{
2x + 2y = 10
3x - 5y = 7
4
Use elimination to solve the system. I'm sorry for threatening you. I didn't mean it.
{
2x + 2y = 10
3x - 5y = 7
Assignment
3.2
page 169
#9 - 20
5
3.2 More elimination
Warm‐ up:
Solve this system for x and y.
(2/3)x ‐ 3y = (1/5)
2x ‐ 9y = 4
No Solution
Infinite Solutions
No Solution
Infinite Solutions
Review for tomorrows quiz:
(1,1)
p204#1-4, 7-10, 13-20 and p129#16-17
Infinite Solutions
6
3.3 Linear Inequalities in
Two Variables
A) Graph the equation as a line you would
without the inequalities
B)
3.3 Linear Inequalities in
Two Variables
3.3 Workbook for Practice.
Due tomorrow.
7
Mr. Brager's job as an X-ray technician
Mr. Brager's job as a
Yak farmer
Notes so far:
- Mr. Brager doesn't lie
- He had jobs this summer as an X-ray
technician and a Yak farmer
15x + 20y ≥ 400
X-ray
Yak farm
$15 per hour
$20 per hour
How many hours would Mr. Brager have to work at
each job to make at least $400 a week?
Graph the linear inequality.
2y - 3 ≤ 4x
8
Graph the linear inequality.
Graph the linear inequality.
2y - 3 ≤ 4x
6x + 3y > 12
Graph the linear inequality.
Assignment
6x + 3y > 12
3.3
page 176
#16 - 22
3.3 Linear Inequalities in Two Variables
Graph the linear inequality.
y ≤ -1
Day 2
Graph the linear inequality.
x>4
9
Graph the linear inequality.
Graph the linear inequality.
2x > 2y +3
-5x - 2y > 4
3.4 Systems of Linear Inequalities
Graph the linear inequality.
3x - 4y ≥ 4
A system of linear inequalities is a collection of linear
inequalities in the same variables.
The solution is any ordered pair that satisfies each
of the inequalities in the system.
Graph the system.
Describe all points that lie
either in the shaded region
or on its boundary.
{
x≥0
y≥0
y > -2x + 5
y ≤ 3x + 1
10
Graph the system.
Graph the system.
{
y ≥ -x - 1
y ≤ 2x + 1
x<1
Graph the system.
{
{
y ≥ -x - 1
y ≤ 2x + 1
x<1
Graph the system.
y > -x - 2
y>x+3
y≤3
{
y > -x - 2
y>x+3
y≤3
3.4 Systems of Linear Inequalities
Assignment
3.4
page 183
#9 - 16
3x-2y 4
x+y 4
x-y 7
x 0
y 0
11
Warm-Up
1.
3.
Write the system of inequalities
{
y ≥ 2x - 1
x>1
Graph the system of inequalities.
{
2y < 2x - 2
y + 2x < 3
y ≥ -1
{
2. x < ­1
y ≥ 2
Graph the system of inequalities.
{
y + 2x ≥ 0
3y ≥ 6x - 12
y<3
Graph the inequality.
-3 ≤ x ≤ 2
Graph the inequality.
Assignment
-1 < y < 3
3.4
pages 183 - 185
#21-25, 33-34, 37-42, 48, 50
12
3.5 Linear Programing Day 1
Warm Up:
1) Graph, shade and find the verticies (points)
of intersection.
y -2
y 2x
y -2x + 4
3.5 Linear Programing Day 1
Warm Up:
1) Graph, shade and find the points of
intersection.
y -2
y 2x
y -2x + 4
Graph the feasible region for each set
of constraints
10)
Linear programming is a method used to find
optimal solutions for problems that have
more than one inequality.
The inequalities are called the constraints
The solution (shaded region) is called the
feasible region
You will also have an objective function
where you will be substituting in values for x
and y to Maximize or minimize the problem.
3.5 #10-21 Solutions
x + 2y 8
2x + y 10
x 0
y 0
13
3.5 Day 2 of Linear programming
Today we will focus more on using the
"objective Function" to find the maximum
and minimum values.
Example: Find the maximum and
minimum values, if they exist, of
each objective function for the
given constraints
P = 2y + 3x
Constraints:
1﴿ Graph the constraints
2﴿Find the verticies of the feasible region.
3﴿ Substitute the verticies into the objective function
x + 2y 16
-3x + 4 y 12
x 0
y 0
3.5 Linear Programming Day 3
Our goal today is to still find the maximum
and minimum values but today we will read
a word problem and create the constraints
and objective function.
Practice 3.5 #32,33,36,37
14
15