Chapter 5 Test: Algebra 2

Elimination

(Addition method) Getting rid of one variable and solving for the other Example: 2x -y = -12 4x + y = 24 Eliminate you Y terms Solve for X 6x = 12 X = 2 Plug 2 in for X on an equation, solve for Y

Parallel

No solitons System: Inconsistent Equations: Independent

Coincident

An infinite number of solutions Systems: Consistent Equations: Dependent

Consistent system

At least one solution

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Do not satisfy the equation Dashed lines on the graph ---------------

Independent

Equations they are not on the same line

Elementary Row Operations

Ero 1: Interchange two equations Ero 2: Multiply one equation by a non zero constant Ero 3: Add a multiple of one equation to another equation, replacing the second equation Example: -4x + 5y -5z= -6 Ero 1 3x + 2y + 2z = 16 Ero 2 -4x+ 3y + z = -10 Ero 3 (Ero 1 --- Ero 2) Interchange Ero 1 and Ero 2 3x +2y+ 2z = 16 -4x + 2y+ 2z = -6 -4x + 3y + z = -10

System of Equations

Example: x + 3y= 3 2x -y = -1

Graphing two variable inequality

Figure out two points they will satisfy the equation Example: 2x -3y < 6 (0,0) 2(0) - 3(0) < 6 0 -0 < 6 0< 6 true

Constraints

Limiting factors A higher demand for mountain bikes Sells 200 mountain bikes and only can sell 100 road bikes

Inconsistent

Multiple solutions

Find the maximum and minimum values of the objective function

Objective: Z= 2x - 4y Constraints: 6x -y > -5 X+ y < 5 X - 6y < 5 Vertex: (0,5). 2(0) -4(5) = -20 Vertex: (-1,-1) 2(-1) - 4(-1) = 2 Vertex: (5,0) 2(5) - 4(0) = 10 Maximum Z: -20 Minimum Z: 10

Intersecting Lines

One solution System: Consistent Equations: Independent

Substitution

One variable eliminates another, by substituting one equations value for another Example: 3x + y = 5 3x - y = 7 Y= -3x +5 Isolate the y variable Substitute the remaining expression into the other equation 3x + 3(-3x+ 5) =7 3x -9x + 15 = 7 -6x = -8 Divide by 6 X = 4/3 Solve one equation for Y by plugging in 4/3 for X Y= 1

Linear programming

Problems that assist in in decision- making

Objective Function

Represents profits Andy's bike shops $50 profit in Mountain bikes, $75 profit from road bikes X= mountain bikes Y= road bikes P = 50x + 75y

Greater than, less than or equal to

Satisfies the equation Solid line on the graph ————-———

Dependent

Share the same line

Three Variables Linear Equations

Solve for X, Y and Z Using the elimination method X - 3y + 2z = 5 (Equ 1) 2x -4y+ 3z = 8 (Equ 2) -2x + 4y -2z = -12 (Equ3)

Feasible solution region

The intersection of inequalities in the system Example: X less than or equal to 200 Y less than or equal to 100 Cost of a mountain bike is $400 Cost of a road bike is $800 400x + 800y less than or equal to 120,000

Optimal objective value

The maximum profit value between two items Example: If Andy sold 100 mountain bikes and 80 road bikes 50(100) + 75(80) = 11,000 Sells at maximum demand 50(200) + 75(100) = 17,500

Intersection

The region that is overlapped by two points meeting at a single point

Vertex

The solution region