movie ticket and popcorn

Question

One family spent $34 for five tickets and three boxes of popcorn. A second family bought six tickets and two boxes of popcorn for $36. Find the cost of one movie tickets and the cost of one box of popcorn

Answer 1

Let x = tickets and let y = popcorn:

Set up the equations:

Family 1: 5x + 3y = 34

Family 2: 6x + 2y = 36

 

Substitution will work better here.

Arrange one equation to isolate a variable.

2y = -6x + 36

y = -3x + 18

 

Substitute in value for y in other equation:

5x + 3(-3x + 18) = 34

5x - 9x + 54 = 34

-4x = -20

x = 5

Substitute x value in for one of the equations:

5(5) + 3y = 34

25 + 3y = 34

3y = 34 - 25

3y = 9 

y = 3

Substitute both values in to check for truth in both equations:

5(5) + 3(3) = 34

25 + 9 = 34

34 = 34 ; check

6(5) + 2(3) = 36

30 + 6 = 36

36 = 36 ; check

 

Tickets are 5 dollars each and each box of popcorn is 3 dollars.

Answer 2

Assign variables to the two things you need to find.  I will use x for the ticket price and y for the popcorn price.  From the sentences, build the equations.

34 = 5x + 3y

6x + 2y = 36

 

That is your system.  You can choose to solve it by elimination or by substitution.  Looking at the equations, the second equation seems simpler to solve for one variable.  I will have to either divide the 6 and the 36 by 2, or divide the 2 and the 36 by 6.  I choose to do the first.  Solve the second equation by y.

 

6x + 2y = 36 (subtract 6x from both sides)

2y = 36 - 6x (divide everyone by 2)

y = 18 - 3x

 

Now return to the first equation and substitute the y term.

5x + 3y = 34 (substitute 18-3x for y)

5x + 3(18-3x) = 34 (distribute the 3)

5x + 54 - 9x = 34 (combine like terms)

54 - 4x = 34 (subtract 54 from both sides)

- 4x = -20 (divide both sides by -4)

x = 5

 

Use that x to find y.

y = 18 - 3(5) = 18-15 = 3.

 

So, the ticket prices (x) were $5 and the popcorn price (y) was $3.

 

Answer 3

Let t be the price of a ticket and p be the cost of a box of popcorn.

34 = 5t + 3p

36 = 6t + 2p

2p = 36-6t

p = 18 - 3t

Substitute that into the first equation.

34 = 5t + 3(18-3t)

34 =5t + 54 - 9t

34 = 54 - 4t

4t = 54 - 34 = 20

t = $5

p = 18 - 3(5) = 3

 

Answer 4

• Define variables:  
• Let's use T for the price of a ticket, and P for price of the popcorn
• Write equations:  
• 5T+3P=34
• 6T+2P=36
• Choose a method to solve these.  I like "elimination" for this one, although it requires changing both equations.  (You could also use substitution here, by first putting one of the equations in the T= or P= form, and substituting it into the other equation.)
• 10T+6P=68 (I doubled this equation so I could get the P's to 6)
• 18T+6P=108 (I tripled this equation to get that same 6 for P)
• 8T=40 (subtract the first equation from the second)
• T=5 (divide both sides by 8)
• Once you have one variable, you can go back in either equation to get the other one
• 5(5)+3P=34 (I randomly chose the first)
• 25+3P=34
• 3P=9
• P=3
• Solution:  Tickets are $5.00 and popcorn is $3.00

 

 

CHECK:

• 5(5)+3(3)=34
• 6(5)+2(3)=36

Answer 5

Hi Meg,

 

Let x = cost of ticket

Let y = cost of popcorn

 

now make two equations (because you need two equations when you have two unknowns

 

5x + 3y = 34  (5 tickets and 3 popcorn = 34

6x + 2y = 36

 

Now multiply the first equation by 2 and the second by 3  (this will make the y values have the same coefficient

 

10x + 6y = 68

18x + 6y = 108

 

now subtract

 

-8x = -40

 

x = 5   (price of ticket

 

now go back and solve for y

 

5x + 3y = 34

25 + 3y = 34

 

3y = 9

y = 3       ( price of popcorn

 

Hope this helps