L ESSON 2.12 U SING S YSTEMS OF E QUATIONS & I NEQUALITIES IN THE R EAL W ORLD Concept: Represent and Solve Systems EQ: How are systems of inequalities and equations used in the real world? (Standards REI.5-6,10-11) Vocabulary : System of Equations, System of Inequalities, Event

S TEPS TO WRITING A SYSTEM OF EQUATIONS OR INEQUALITIES FROM A WORD PROBLEM 1. Read the problem. 2. Make a table or a list of what you know. 3. Identify the two unknown quantities and define each of them with a variable. 4. Exam your table or list to identify what two events are being compared. 5. Write two equations or inequalities (one per event ) relating the known information to the two variables. 6. Solve the system. 7. Interpret and apply your solution to the context of the problem.

B OOMERANGS

G UIDED P RACTICE – E XAMPLE 1 Phil and Cathy plan to make and sell boomerangs for a school event. The money they raise will go to charity. They plan to make them in two sizes: small and large. Phil will carve them from wood. The small boomerang takes 2 hours to carve, while the large one takes 3 hours to carve. Phil has a total of 24 hours available for carving. Cathy will decorate them. She only has time to decorate 10 boomerangs of either size. Write and solve a system of equations to determine how many small and large boomerangs they should make?

Two size boomerangs: small and large Small boomerang takes 2 hours to carve Large boomerang takes 3 hours to carve Phil has a total of 24 hours to carve Cathy only has time to decorate 10 boomerangs of either size. # of small boomerangs to make # of large boomerangs to make S = # of small boomerangs to make L = # of large boomerangs to make Carving Decorating

G UIDED P RACTICE – E XAMPLE 1, CONTINUED Solve the system of equations using the substitution or elimination method.

Two size boomerangs: small and large Small boomerang takes 2 hours to carve Large boomerang takes 3 hours to carve Phil has a total of 24 hours to carve Cathy only has time to decorate 10 boomerangs of either size. # of small boomerangs to make # of large boomerangs to make S = # of small boomerangs to make L = # of large boomerangs to make Carving Decorating

G UIDED P RACTICE – E XAMPLE 2 An artist wants to analyze the time that he spends creating his art. He makes oil paintings and watercolor paintings. The artist takes 8 hours to paint an oil painting. He takes 6 hours to paint a watercolor painting. He has set aside a maximum of 24 hours per week to paint his paintings. The artist then takes 2 hours to frame and put the final touches on his oil paintings. He takes 3 hours to frame and put the final touches on his watercolor paintings. He has set aside a maximum of 12 hours per week for framing and final touch-ups. Write a system of inequalities that represents the time the artist has to complete his paintings. Graph the solution.

Oil paintings Watercolor paintings 8 hours to paint oil paintings 6 hours to paint watercolor paintings 24 hours maximum to paint paintings 2 hours to frame and put final touches on oil paintings 3 hours to frame and put final touches on watercolor paintings 12 maximum to frame and on put final touches # of oil paintings the artist makes # of watercolor paintings the artist makes x - # of oil paintings the artist makes y - # of watercolor paintings the artist makes Painting Framing & putting on final touches

Graph both inequalities on the same coordinate plane. First get them in slope-intercept form. G UIDED P RACTICE – E XAMPLE 2, CONTINUED

Now, think about what must always be true of creating the paintings: there will never be negative paintings. This means the solution lies in the first quadrant.

G UIDED P RACTICE – E XAMPLE 2, CONTINUED The solution is the darker shaded region; any points that lie within it are solutions to the system. The point (1, 1) is a solution because it satisfies both inequalities. The artist can create 1 oil painting and 1 watercolor painting given the time constraints he has. Or, he can create no oil paintings and 4 watercolor paintings, (0, 4). However, he cannot create 4 oil paintings and 1 watercolor painting, because the point (4, 1) only satisfies one inequality and does not lie in the darker shaded region.

Oil paintings Watercolor paintings 8 hours to paint oil paintings 6 hours to paint watercolor paintings 24 hours maximum to paint paintings 2 hours to frame and put final touches on oil paintings 3 hours to frame and put final touches on watercolor paintings 12 maximum to frame and on put final touches # of oil paintings the artist makes # of watercolor paintings the artist makes x - # of oil paintings the artist makes y - # of watercolor paintings the artist makes Painting Framing & putting on final touches Any points that lie within the darker shaded region. The point (1,1) is a solution.

P ARTNER P RACTICE – E XAMPLE 3 Movie tickets are $9.00 for adults and $5.00 for children. One evening, the theater sold 45 tickets worth $ Write and solve a system of equations to determine how many adult tickets were sold and how many children’s tickets were sold.

P ARTNER P RACTICE – E XAMPLE 4 Esther has a total of 23 dimes and pennies. The value of her coins is $1.85. Write and solve a system of equations to determine how many dimes and how many pennies Esther has.