Math 1
Linear Programming: Equations and Vertices
Unit 2 Day 7
Linear programming: a method for finding a __________________________________________ of some quantity, given a set of constraints. Some real
world problems involve multiple linear relationships. Linear programming helps us to find a ___________________to these problems. In a linear
programming problem, you are given a _________________________________, which contains all the points that satisfy the constraints. The quantity you
are trying to maximize or minimize if given by the __________________________. Often this quantity is cost or profit.
Find the coordinates of the vertices of the figure formed by….
Y 1 : y  1
Y 1 : x  y  1
Example 1:
Y2: x  y  6
Y 3 : 12 y  6 x  36
Example 2:
Solve for y….
Solve for y….
Y1:
Y1:
Y2:
Y2:
Y3:
Y3:
Vertices of the Figure:
1. Graph each line on a coordinate plane
2. Find the intersections of the lines
3. Determine the coordinates of those intersections
Y 2 : 2x  y  5
Y 3 : x  3
Vertices of the Figure:
Intersection of Y1 and Y2:_____________
Intersection of Y1 and Y2:_____________
Intersection of Y1 and Y3:_____________
Intersection of Y1 and Y3:_____________
Intersection of Y2 and Y3:_____________
Intersection of Y2 and Y3:_____________
Linear Programming Procedure:
1) Find vertices.
2) Plug vertices into function to be maximized or minimized.
3) Select the greatest result to be the “maximum,” and the least result to
be the “minimum.”
Vertices
Objective Function
P=x+y
Solution to function
f(x,y)
vertex
Plug x from vertex into x in
function and same for y
Plug x from vertex into x in
function and same for y
Plug x from vertex into x in
function and same for y
Choose max/min
values from here
Choose max/min
values from here
Choose max/min
values from here
vertex
vertex
Example 1: Graph the following system of inequalities. Name the
coordinates of the vertices of the feasible region. Find the maximum
and minimum values of the objective function
for this
Example 2: Graph the following system of inequalities. Name the
coordinates of the vertices of the feasible region. Find the maximum
and minimum values of the objective function
for this
region:
region:
x5
y4
x y  2
Solve for y….
Y1:
Solve for y….
Y1:
Y2:
Y2:
Y3:
Y3:
Y4:
Vertices of the Figure:
Vertices of the Figure:
Intersection 1:_____________
Intersection 1:_________
Intersection 2:_________
Intersection 2:_____________
Intersection 3:_________
Intersection 4:_________
Intersection 3:_____________
Substitute in to Objective Function:
Substitute in to Objective Function:
Vertex 1: ____________________________________
Vertex 1: ______________________________
Vertex 2: ____________________________________
Vertex 2: ______________________________
Vertex 3: ____________________________________
Vertex 3: ______________________________
Vertex 4: ____________________________________
Maximum: ___________ Minimum: ___________
Maximum: ______________ Minimum: ______________