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 graphingx – 2y = 0x + y = 6

2.1 Solve by Graphing

ConsistentSystem that has at least one solution

InconsistentNo solution

IndependentExactly one solution

DependentMany solutions

In calculator

Enter both equations in slope intercept form under the “y=“ button

2nd calc5: intersectSelect 1st curveSelect 2nd curve“Guess?” press enter

Intersecting Lines

One solutionConsistent and independent

Same Line

Infinitely Many SolutionsConsistent and Dependent

9x – 6y = -66x – 4y = -4

Parallel Lines

No SolutionSame slope -- never intersectInconsistent

15x – 6y = 05x – 2y = 10

Substitution Method1. Solve for one variable in terms of the

other2. Substitute into other equation3. Solve4. Plug in for other variable

x + 4y = 26x – 5y = -10

Elimination Method

Multiply one or both equations by certain numbers to create opposites

Add equations to eliminate a variableSolve for the remaining variablePlug back in to find the other

Examples

x + 2y = 10x + y = 6

2x + 3y = 125x – 2y = 11

-3x + 5y = 126x – 10y = -21

2.2 Systems of 3 Variables

Ordered Triple - (x, y, z)1. Eliminate one variable2. Solve system of two equations3. Plug in two values to find third

Ex 5x + 3y + 2z = 22x + y – z = 5x + 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 = 5x + 2y – 4z = 76x + 3y - 9z = 15

No Solution

Yields false result

3x – y – 2z = 46x + 4y + 8z = 119x + 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 graphs3. This is the “double” shaded region

2.3

2.6 Solving Systems of Inequalities

ExamplesY>x+1

│y │≤3y≥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 usedFeasible Region – Intersection of the graphsBounded – When the graph of a system of

constraints is a polygonal regionUnbounded – when a system of inequalities

forms a region that is openVertices – 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≥1y ≥02x+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≤2X-2y≤4X+y≥-2

2.7 Linear ProgrammingWord Problem Procedure

Define the variablesWrite a system of inequalitiesGraph systemFind the coordinates of the vertices of the feasible

regionWrite a function to be maximized or minimizedSubstitute the coordinates of the verticies into the

functionSelect 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.