Set up the System
A group of trick-or-treaters pooled their candy. They have three types of candy: chocolate bars, lollipops, and candy corn bags. Cory, Josh, and Dan each bought a specific combination of these treats.
3x + 4y + 8z = 36.65
5x + 3y + 10z = 37.5
4x + 5y + 6z = 43.45
5x - 3y = -8
y = 2x
(8, 16)
x + y = 24
x - y = 6
(15, 9)
At a jewelry store, Finn buys 3 necklaces, 1 ring, and 2 bracelets for $17. Rosy buys 1 necklace, 4 rings, and 5 bracelets for $31. Mia buys 6 necklaces, 2 rings, and 1 bracelet for $19.
SOLUTION: (2, 1, 5)
What is the meaning of this solution?
Necklaces cost $2.
Rings cost $1.
Bracelets cost $5.
A football team scored a total of 50 points in one game. They scored 14 times. The scoring was made up of touchdowns (6 points each), PATs (1 point each) and field goals (3 points each). They had three more touchdowns that field goals.
x + y + z = 14
6x + y + 3z = 50
x = z + 3
x = 2y - 24
x = -19 + 7y
(-26, -1)
2x - 5y = -26
3x + 5y = 36
(2, 6)
Rosie is solving a system of 3-variable equations to find the number of adult, child, and senior tickets she will need to sell for her fundraiser. Which of the following is NOT a reasonable result?
A: (250, 250, 250)
B: (500, 250, 0)
C: (700, 100, -50)
D: (1000, 900, 800)
C (You cannot sell a negative number of tickets.)
The new animated feature film is now playing three times a day at the local theater. One day, there were 20 adults, 43 children, and 10 seniors and the thater made $614 in ticket sales. At the next showing, there were 24 adults, 59 children, and 20 seniors with the theater making $852 in ticket sales. At the last showing, the theater made $405 with 13 adults, 30 children, and 5 seniors.
20x + 43y + 10z = 614
24x + 59y + 20z = 852
13x + 30y + 5z = 405
3x + 4y = 12
x = 2/3y - 2
(0, 3)
6x + 4y = 42
6x - y = -3
(1, 9)
Beckett solved this problem: A football team scored a total of 50 points in one game. They scored 14 times. The scoring was made up of touchdowns (6 points each), PATs (1 point each) and field goals (3 points each). They had three more touchdowns that field goals.
Here is his solution: (6, 5, 3). Is he correct?
YES!
(6 touchdowns, 5 PATs, and 3 field goals make 14 scores for 50 points, with 3 more TDs than FGs.)
At a bake sale in October, a group sells cookies, brownies, and mini-apple pies.
15x + 12y + 8z = 70
10x + 12y + 4z = 48
x + z = 6
2x + 10y = 38
x = y - 5
(-1, 4)
x - 3y = -18
3x + 7y = 26
(-3, 5)
A school bake sale sold a total of 140 brownies, cookies, and cupcakes. They sold twice as many cookies as cupcakes. If brownies cost $2.50 each, cookies cost $2 each, and cupcakes cost $3 each, the total revenue was $335. How many of each item were sold?
Is (52, 60, 28) a reasonable solution?
No! (While the number do add up to a total of 140, 60 is not twice as much as 28, and the total cost would be $329 rather than $335.)
Three friends, Sarah, Liam, and Maya, are figuring out their budget for a weekend of fun. They decide on three activities: a movie, going to the arcade, and getting some ice cream. To save money, they are tracking how much they each spend.
x + 2y + z = 33
2x + 3y + 2z = 40
x = 2y
3x + 5y = 2
y = 9x - 38
(4, -2)
Eliminate ONE variable:
2x + 3y - z = 84
5x - 7y + z = 14
x + y + z = 38
7x - 4y = 98
4x - 8y = -24
Caleb is solving this problem:
The ones digit of a three digit number is 2.5 times greater than the hundreds digit while the tens digit is five more than the hundreds digit. The sum of the three digits is fourteen.
He sets up the system:
x = 2.5z; y = z + 5; x + y + z = 14
and solves to find that x = 5, y = 7, and z = 2.
What is the three digit number?
275