lesson 2.12 using systems of equations & inequalities in the real world
DESCRIPTION
Lesson 2.12 Using 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. - PowerPoint PPT PresentationTRANSCRIPT
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
STEPS 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.
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?
• 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 makeL = # of large boomerangs to make
• Carving• Decorating
GUIDED PRACTICE – EXAMPLE 1, CONTINUED
Solve the system of equations using the substitution or elimination method.
Substitution MethodSolve the second equation for s.
Substitute this new equation intoThe first one.
Substitute this value into the amendedEquation from step 1.
• 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 makeL = # of large boomerangs to make
• Carving• Decorating
They should make 4 large and 6 small boomerangs.
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.
• 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 makesy - # 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.
6 𝑦 ≤−8 𝑥+24 𝑦 ≤−43𝑥+4
3 𝑦≤−2𝑥+12 𝑦 ≤−23𝑥+4
GUIDED PRACTICE – EXAMPLE 2, CONTINUED
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.
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.
• 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 makesy - # of watercolor paintings the artist makes
• Painting• Framing & putting on final touches
Any points that lie within the darkershaded region. The point (1,1) is a solution.
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.
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.
