Chapter 2 Systems of Linear Equations and Inequalities

2.1 Solve by Graphing System of equations - Two or more equations with the same variables Solutions - Where the lines intersect Solve the systems of equations by graphing x – 2y = 0 x + y = 6

2.1 Solve by Graphing Consistent System that has at least one solution Inconsistent No solution Independent Exactly one solution Dependent Many solutions

In calculator Enter both equations in slope intercept form under the “y=“ button 2 nd calc 5: intersect Select 1 st curve Select 2 nd curve “Guess?” press enter

Intersecting Lines One solution Consistent and independent

Same Line Infinitely Many Solutions Consistent and Dependent 9x – 6y = -6 6x – 4y = -4

Parallel Lines No Solution Same slope -- never intersect Inconsistent 15x – 6y = 0 5x – 2y = 10

Substitution Method 1. Solve for one variable in terms of the other 2. Substitute into other equation 3. Solve 4. Plug in for other variable x + 4y = 26 x – 5y = -10

Elimination Method Multiply one or both equations by certain numbers to create opposites Add equations to eliminate a variable Solve for the remaining variable Plug back in to find the other

Examples x + 2y = 10 x + y = 6 2x + 3y = 12 5x – 2y = 11 -3x + 5y = 12 6x – 10y = -21

2.2 Systems of 3 Variables Ordered Triple - (x, y, z) 1. Eliminate one variable 2. Solve system of two equations 3. Plug in two values to find third Ex5x + 3y + 2z = 2 2x + y – z = 5 x + 4y + 2z = 16

There are 49,000 seats in a sports stadium. Tickets for the seats in the upper level sell for $25, the ones in the middle level cost $30, and the ones in the bottom level cost $35 each. The number of seats in the middle and bottom levels together equal the number of seats in the upper level. When all of the seats are sold for an event, the total revenue is $1,419,500. How many seats are in each level?

Infinite Solutions Yields a true result 2x + y - 3z = 5 x + 2y – 4z = 7 6x + 3y - 9z = 15

No Solution Yields false result 3x – y – 2z = 4 6x + 4y + 8z = 11 9x + 6y + 12z = -3

2.6 Solving Systems of Inequalities Solutions - All ordered pairs that satisfy the system 1. Graph each inequality (under “y=“) 2. Solution is intersection of both graphs 3. This is the “double” shaded region

2.3

2.6 Solving Systems of Inequalities Examples Y>x+1 │y │≤3 y≥2x-3 y<-x+2

When NASA chose the first astronauts in 1959, size was an important issue because the space available inside the Mercury capsule was very limited. NASA wanted men who were at least 5’4”, but no more than 5’11”, and who were between 21 and 40 years of age. Write and graph a system of inequalities that represents the range of heights and ages for qualifying astronauts.

2.6 Solving Systems of Inequalities Find the coordinates of the vertices of the figure formed by x+y≥-1, x-y≤6, and 12y+x ≤32. Find the coordinates of the vertices of the figure formed by 2x-y≥-1. x+y ≤4, and x+4y ≥4.

2.7 Linear Programming Constraints – Inequalities used Feasible Region – Intersection of the graphs Bounded – When the graph of a system of constraints is a polygonal region Unbounded – when a system of inequalities forms a region that is open Vertices – where the maximum or minimum value of a related function always occurs

2.7 Linear Programming Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function F(x,y)=3x+y for this region. x≥1 y ≥0 2x+y≤6

2.7 Linear Programming Find the maximum and minimum values of the function f(x,y)=2x+3y for the unbounded region. -x+2y≤2 X-2y≤4 X+y≥-2

2.7 Linear Programming Word Problem Procedure Define the variables Write a system of inequalities Graph system Find the coordinates of the vertices of the feasible region Write a function to be maximized or minimized Substitute the coordinates of the verticies into the function Select the greatest of least result and answer the problem

As a receptionist for a vet, one of the tasks is to schedule appointments. She allots 20 minutes for a routine office visit and 40 minutes for a surgery. The vet cannot do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visit costs $55 and most surgeries cost $125, find a combination of office visits and surgeries that will maximize the income the vet practice receives per day.