Solving by Graphing (or substitution)
y = 2x - 2
y = 1/2x - 2
(0,-2)
y = 2x + 3
y = 5x
(1,5)
x + 3y = -2
x - 3y = 16
(7,-3)
Without graphing or solving, determine the number of solutions to the system. Explain your reasoning.
y = 5x - 9
y = 5x + 9
-----
5x - 9 = 5x + 9
-9 = 9
no solution; the lines are parallel
Solves to nonsense, _ does NOT equal _
Your family is buying tickets for an amusement park. Child tickets are $20 each, and adult tickets are $50. Your family buys a total of 10 tickets and spends $320.
How many of each ticket you purchase?
Write the equations for x children tickets and y adult tickets and solve.
x + y = 10
2x + 50y = 320
4 adult tickets and 6 child tickets
y = x + 1
y = 4x + 7
*or substitution - set them equal*
(-2,-1)
y = 2x - 4
7x - 2y = 5
(-1,-6)
2x - y = 9
4x + y = 21
(5,1)
Without graphing or solving, determine the number of solutions to the system. Explain your reasoning.
y = 6x + 2
y = 3x + 11
------
6x + 2 = 3x + 11
3x = 9
x = 3
y = 20
(3,20)
1 solution; Different slopes
Solves to a point (x, y)
You have a total of 42 math and science problems for homework. You have 10 more math problems than science problems.
How many problems do you have in each subject?
Write the equations for x math problems and y science problems and solve.
x + y = 42
x = y + 10
26 math problems
16 science problems
y = x - 5
y = 3x -1
*or substitution, set them equal*
(-2,-7)
4x + 2y = 0
y = 1/2x -5
(2,-4)
5x - 2y = -13
5x + y = -1
*Watch your signs*
(-1,4)
Without graphing or solving, determine the number of solutions to the system. Explain your reasoning.
y = 8x - 2
y + 8x = -2
------
8x - 2 + 8x = -2
16x = 0
x = 0
y = -2
(0, -2)
1 solution; different slopes
Solves to one point (x, y)
You are planning a party and buy a total of 50 hamburgers and hotdogs for $90. You pay $2 per hamburger and $1.50 per hotdog.
How many of each did you buy?
Write the equations for x hambugers and y hotdots and solve.
x + y = 50
2x + 1.50y = 90
30 hamburgers and 20 hotdogs
y = -4x - 7
y = -x + 2
*or substitution, set them equal*
(-3,5)
x = 5y + 3
2x + 4y = -1
(1/2, - 1/2)
-6x + 5y = 25
-2x - 4y =14
(-5,-1)
Without graphing or solving, determine the number of solutions to the system. Explain your reasoning.
4x - 4y= -8
y = x + 2
-----
4x - 4(x + 2) = -8
4x - 4x - 8 = -8
-8 = -8
infinitely many solutions; same equations
Solves to __ equal __ ( same values)
You went to concessions and bought 3 sodas and 2 Chick-Fil-A sandwiches and spent $18. A friend bought 2 sodas and 1 CFA sandwich and spent $10.
How much is each at the concessions stand?
Write the equations for x price of soda and y price of CFA sandwich and solve.
3x + 2y = 18
2x + y = 10
sodas are $2 and CFA is $6.
y = -1/2x -6
y = 1/4x - 9
(4,-8)
2x + 3y = -3
y = 2x - 5
(1.5, -2)
5x + 5y = 15
2x + y = 3
(0,3)
Without graphing or solving, determine the number of solutions to the system. Explain your reasoning.
2x - 4y = 10
-12x + 24y = -60
-----
Multiply top by 12, botton by 2
24x - 48y = 120
-24x + 48y = -120
0 = 0
Infinitely many solutions; same equation
Solves to __ equals __ (same value)
You are comparing prices for a party.
Spot A charges $200 space rental and $5 per person.
Spot B charges $60 for space rental and $25 per person.
Write the equations for each spot and solve for how many people will be the same price.
y = 200 + 5x OR y = 5x + 200
y = 60 + 25x OR y = 25x + 60
at 7 people, both spots cost $235