Systems of Equations
y = 3x + 2
y = -2x -3
(-1, -1)
If 8 pens and 7 pencils cost $3.37 while 5 pens and 11 pencils cost $3.10, how much does each pen and each pencil cost? Define your variables and create a system of equations.
x = cost of pens
y = cost of pencils
8x + 7y = 3.37
5x + 11y = 3.1
Pens cost $0.29 and pencils cost $0.15
y = 3x - 10
6x -2y = 20
Infinitely many solutions
An internet provider charges $25 plus an additional $0.50 per hour of internet you use. Another provider charges $31 plus an additional $0.20 per hour of internet you use. When will they cost the same and how much will they cost? Define your variables and create a system of equations.
c = total cost
h = number of hours of internet
25 + 0.5 h = c
31 + 0.2h = c
After 20 hours they will both cost $35
-3x + 2y = 4
6x -4y = -4
No solution
Loren's marble jar contains plain marbles and colored marbles. If there are 32 more plain marbles than colored marbles and there are 180 marbles in the jar, how many colored marbles and how many plain marbles does Loren have? Define your variables and create a system of equations.
p = number of plain marbles
c = number of colored marbles
c + p = 180
c + 32 = p
She has 74 colored marbles and 106 plain marbles
3x−4y=8
18x−5y=10
(0, -2)
Jorge is having a bake sale. It costs $20 to get the table and it costs $0.50 to make each cookie. Jorge sells the cookies for $1.30 each. When will Jorge make more money then he spent and how much money is it? Define your variables and create a system of equations.
m = amount of money, c = number of cookies
m = 20 + 0.5c
m = 1.3 c
He should sell more than 25 cookies. At 25 cookies, his income and his expenses would both be $32.50
Makenna wants to sell clay dogs and cats to save up money for a new sweatshirt. She sells clay dogs for $5 a piece and clay cats for $4 a piece. She has 8 items to sell. She wants to make at least $30. Create a system of inequalities to represent the situation, graph it, and name a possible solution.
x = number of dogs, y = number of cats
30 ≤ 5x + 4y
x + y ≤ 8
graph
3x = y
y = z + x
y = 2z - 3x +2
(1, 3, 2)
Ann can paddle her canoe 15 mi downstream (with the current) in 2 h. The return trip upstream (against the current) takes 3 times as long. What is the rate of the current and what is Ann's speed? (Remember! d = rt) Define your variables and create a system of equations.
a = the speed Ann can canoe
c = the speed of the current
15 = (a + c) 2
15 = (a - c) 6
Ann's speed is 5 mi/h and the current's speed is 2.5 mi/h
Nick is selling bracelets and earrings at a craft fair. He will sell bracelets for $3 and earrings for $2. He wants to make at least $30. He only has 12 boxes so he can sell at most 12 items. Create a system of inequalities to represent the situation, graph it, and name a possible solution.
x = number of bracelets, y = number of earrings
30 ≤ 3x + 2y
x + y ≤ 12
graph