3 4-1 Systems of Equations in two Variables How do you solve a system of equations in two variables graphically?
4 VocabularySystems of equations: two or more equations using the same variablesLinear systems: each equation has two distinct variables to the first degree.Independent system: one solutionDependent system: many solutions, the same lineInconsistent system: no solution, parallel lines
5 Directions: Solve each equation for y Graph each equation State the point of intersection
19 4-2B and Solving Systems of Equations —LINEAR COMBINATION —ELIMINATION METHOD Consistent and Dependent SystemsHow do you solve a system of equations in two variables by linear combinations?What makes a system dependent, independent, consistent, or inconsistent?
20 Combination/Elimination 1)LOOK FOR OR CREATE A SET OF OPPOSITESA) TO CREATE USE THE COEFFICIENT OF THE 1ST WITH THE SECOND AND VICE VERSAB) MAKE SURE THERE WILL BE ONE + & ONE –2) ADD THE EQUATIONS TOGETHER AND SOLVE3) SUSTITUTE IN EITHER EQUATION AND SOLVE FOR THE REMAINING VARIABLE
28 4-3 Using a system of two Equations How do you translate real life problems into systems of equations?
29 USE ROPES:Read the problemOrganize your thoughts in a chartPlan the equations that will workEvaluate the SolutionSummarize your findings
30 Example:The sum of the first number and a second number is -42. The first number minus the second is 52. Find the numbers1st numberx2nd numberyx y = -42x y = 52
31 Example:Soybean meal is 16% protein and corn meal is 9% protein. How many pounds of each should be mixed together to get a 350 pound mix that is 12% protein?Soybean mealx.16Corn mealy.09x y = 350.16x + .09y = .12 • 350
32 Example:A total of $1150 was invested part at 12% and part at 11%. The total yield was $ How much was invested at each rate?12% investmentx.1211% investmenty.11x y = 1150.12x + .11y =
33 Example:One day a store sold 45 pens. One kind cost $8.75 the other $ In all, $ was earned. How many of each kind were sold?Type 1x8.75Type 2y9.75x y = 458.75x y =
42 4-5 Using a System of Three Equations How do you translate word problems into a system of three equations?
43 Example:The sum of three numbers is The third is 11 less than ten times the second. Twice the first is 7 more than three times the second. Find the numbers.1st numberx2nd numberY3rd numberzx + y + z = 105z = 10y – 112x = y
44 Example: Sawmills A, B, C can produce 7400 board feet of lumber per day. A and B together can produce 4700 board feet, while B and C together can produce 5200 board feet. How many board feet can each mill produce?Mill AxMill ByMill Czx + y + z = 7400x + y = 4700y + z = 5200
57 4-8 Using Linear Programming EQ: What is linear programming?
58 VOCABULARY:Linear programming– identifies minimum or maximum of a given situationConstraints—the linear inequalities that are determined by the problemObjective—the equation that proves the minimum or maximum value.
59 Directions: Read the problem List the constraints List the objective Graph the inequalities finding the feasible regionSolve for the vertices (the points of intersection)Test the vertices in the objective
60 Example:What values of y maximize P given Constraints: y≥3/2x -3 y ≤-x + 7 x≥0 y≥0 Objective: P = 3x +2yxyP
61 You are selling cases of mixed nuts and roasted peanuts You are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. If both sell equally well, how can you maximize the profit assuming you will sell everything that you buy?xyP
62 Partner Problem (sample was #8) A florist has to order roses and carnations for Valentine’s Day. The florist needs to decide how many dozen roses and carnations should be ordered to obtain a maximum profit. Roses: The florist’s cost is $20 per dozen, the profit over cost is $20 per dozen. Carnations: The florist’s cost is $5 per dozen, the profit over cost is $8 per dozen. The florist can order no more than 60 dozen flowers. Based on previous years, a minimum of 20 dozen carnations must be ordered. The florist cannot order more than $450 worth of roses and carnations. Find out how many dozen of each the florist should order to max. profit!CostTotal orderedProfitxyP=20x + 8y
63 Sample of what must be handed in for Partner problem