Lesson 2.12 Using Systems of Equations & Inequalities in the Real World

1. Lesson 2.12Using Systems of Equations & Inequalities in the Real World 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

2. Steps to writing a system of equations or inequalities from a word problem • Read the problem. • Make a table or a list of what you know. • Identify the two unknown quantities and define each of them with a variable. • Exam your table or list to identify what two events are being compared. • Write two equations or inequalities (one per event) relating the known information to the two variables. • Solve the system. • Interpret and apply your solution to the context of the problem.

3. Boomerangs http://www.youtube.com/watch?v=zl10s5xe2-4

4. Guided Practice – Example 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?

5. # of small boomerangs to make # of large boomerangs to 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 • hoursto carve • Cathy only has time • to decorate 10 boomerangs • of either size. S = # of small boomerangs to make L = # of large boomerangs to make • Carving • Decorating

6. Guided Practice – Example 1, continued Substitution Method Solve the second equation for s. Substitute this new equation into The first one. Substitute this value into the amended Equation from step 1. Solve the system of equations using the substitution or elimination method.

7. # of small boomerangs to make # of large boomerangs to 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 • hoursto carve • Cathy only has time • to decorate 10 boomerangs • of either size. S = # of small boomerangs to make L = # of large boomerangs to make They should make 4 large and 6 small boomerangs. • Carving • Decorating

8. Guided Practice – Example 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.

9. 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

10. Guided Practice – Example 2, continued Graph both inequalities on the same coordinate plane. First get them in slope-intercept form.

11. Guided Practice – Example 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.

12. Guided Practice – Example 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.

13. 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 Any points that lie within the darker shaded region. The point (1,1) is a solution. • Painting • Framing & putting • on final touches

14. Partner Practice – Example 3 Movie tickets are $9.00 for adults and $5.00 for children. One evening, the theater sold 45 tickets worth $273.00. Write and solve a system of equations to determine how many adult tickets were sold and how many children’s tickets were sold.

16. Partner Practice – Example 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.