Systems
Two lines intersect at (−2, 6).
What is the solution of their system?
(−2, 6)
The intersection lies on BOTH lines.
A club sells 54 craft kits altogether. Let x = small kits and y = large kits.
Write an equation for the total kit count.
x + y = 54
Both variables count kits.
A group can buy at most 22 total items. Let x and y count the two item types.
Write the inequality for the total-item limit.
x + y ≤ 22
“At most” includes exactly 22.
Each small sign earns $9 profit; each large sign earns $14 profit.
Let x = small signs; y = large signs.
Write the objective function to maximize.
P = 9x + 14y
Multiply each count by its profit and add.
What three-word phrase do people say at the door when asking for Halloween candy?
Trick or treat!
Solve the system.
y = x + 2
y = −x + 10
(4, 6)
6 = 4 + 2 and 6 = −4 + 10.
Two memberships cost A = 40 + 6v and B = 16 + 9v dollars.
At how many visits do they cost the same? Give the shared cost too.
8 visits; both cost $88.
40 + 6(8) = 88
16 + 9(8) = 88
How do you graph y ≥ −3x + 4?
State the boundary type and which side to shade.
Solid boundary; shade ABOVE the line.
Equality includes points on the boundary.
Profit is P = 8x + 13y dollars.
What is the profit at (4, 3)?
$71
8(4) + 13(3) = 32 + 39 = 71
What is a pumpkin carved with a face and lit from inside commonly called?
A jack-o’-lantern
How many solutions? Explain.
3x − y = 9
6x − 2y = 4
No solutions.
The lines are y = 3x − 9 and y = 3x − 2.
Same slope, different intercepts.
Adult tickets cost $12; student tickets cost $7. A theater sells 46 tickets for $447.
How many of EACH type were sold?
25 adult tickets and 21 student tickets
25 + 21 = 46
12(25) + 7(21) = 447
Is (3, 4) feasible? Check EVERY constraint.
x ≥ 2
y ≥ 1
2x + y ≤ 11
Yes.
3 ≥ 2; 4 ≥ 1; 2(3) + 4 = 10 ≤ 11.
All three constraints are satisfied.
Feasible corners: (0, 0), (9, 0), (6, 4), (0, 7).
Profit: P = 10x + 18y.
Find the best corner and maximum profit.
(6, 4); maximum profit $132
Corner profits: $0, $90, $132, $126.
In It’s the Great Pumpkin, Charlie Brown, which character waits in the pumpkin patch for the Great Pumpkin?
Linus
A student says this system has only one solution, (0, 4). Is that correct? Explain.
2x + 3y = 12
4x + 6y = 24
No. There are infinitely many solutions.
The second equation is twice the first.
(0, 4) is one point on their shared line.
Mix 20% solution and 60% solution to make 15 liters of 36% solution.
How many liters of EACH should be used?
9 L of 20% solution; 6 L of 60% solution
9 + 6 = 15
0.20(9) + 0.60(6) = 5.4 = 0.36(15)
Small bags cost $4; large bags cost $7. Spend at most $84 and buy at least 12 bags.
Let x = small bags; y = large bags.
Write BOTH limits and the count restrictions.
4x + 7y ≤ 84
x + y ≥ 12
x ≥ 0 and y ≥ 0; x and y are whole numbers.
Feasible corners: (2, 5), (6, 2), (8, 6), (3, 9).
Cost: C = 7x + 9y dollars.
Find the best corner and MINIMUM cost.
(2, 5); minimum cost $59
Corner costs: $59, $60, $110, $102.
The Sanderson sisters are the witches in which Halloween movie?
Hocus Pocus
Accept Hocus Pocus 2 as well.
Find k so the system has NO solution. Explain why your value works.
y = −2x + 7
y = kx − 5
k = −2
Equal slopes and different intercepts produce parallel, distinct lines.
Supply: q = 5p + 30
Demand: q = −7p + 318
p = price in dollars; q = number of items.
Find BOTH the equilibrium price and quantity.
Price: $24; quantity: 150 items
5p + 30 = −7p + 318 gives p = 24.
Both models then give q = 150.
x and y are nonnegative whole numbers.
3x + 4y ≤ 43 and x + y ≤ 11.
If x = 5, what is the greatest possible y? Explain why a larger y will not work.
y = 6
The limits become y ≤ 7 and y ≤ 6.
(5, 6) works; y = 7 violates x + y ≤ 11.
Does (6, 6) maximize P = 18x + 6y?
Feasible corners: (0, 0), (10, 0), (6, 6), (0, 8).
Find the true best corner and profit.
(10, 0); maximum profit $180
Corner profits: $0, $180, $144, $48.
Using every resource does not guarantee most profit.
Before pumpkin lanterns became popular in the United States, what root vegetable was traditionally carved into lanterns in Ireland?
A turnip
FINAL JEOPARDY — 4 minutes
A club makes x posters and y banners.
At most 18 items; at most 26 work-hours.
Each poster takes 2 hours; each banner takes 1.
Make at most 14 banners. Profit: $5 per poster, $8 per banner.
Find the whole-number plan with maximum profit. State the profit and justify your choice.
4 posters and 14 banners; maximum profit $132.
Limits: x + y ≤ 18; 2x + y ≤ 26; y ≤ 14; x, y ≥ 0, whole numbers.
Corners: (0,0), (13,0), (8,10), (4,14), (0,14).
Profits: $0, $65, $120, $132, $112.
The best plan uses 18 items and 22 of 26 hours.
Teacher: adjust each team score by its wager, not this tile’s point value.