Write and solve a system of equations for each situation. Check your answers.
13. Your school sells tickets for its winner concert. Student tickets are $5 and adult tickets are $10. If your school sells 85 tickets and makes $600, how many of each ticket did they sell?
14. A grocery store has small bags of apples for $5 and large bags of apples for $8. If you buy 6 bags and spend $45, how many of each size bag did you buy?
13.
number of adult tickets sold = x
number of student tickets = 85-x
solve: 10x + 8(85-x) = 600
14. set it up the same way as #13
number of bags of small apples = x
number of bags of large apples = 6-x
etc.
20
x+65
A grocery store has small bags of apples for $5 and large bags of apples for $8. If you
buy 6 bags and spend $45, how many of each size bag did you buy?
To solve these systems of equations, we will assign variables to represent the unknown quantities. Let's use the following variables:
Let x represent the number of student tickets.
Let y represent the number of adult tickets.
Let's solve each situation step by step:
13. The school sells student tickets for $5 and adult tickets for $10. The total number of tickets sold is 85, and the total revenue is $600. We can set up the following system of equations:
Equation 1: x + y = 85 (Total number of tickets sold)
Equation 2: 5x + 10y = 600 (Total revenue earned)
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution:
From Equation 1, we can rewrite it as x = 85 - y.
Substituting x = 85 - y into Equation 2, we get:
5(85 - y) + 10y = 600
Simplifying the equation:
425 - 5y + 10y = 600
Combining like terms:
5y = 600 - 425
5y = 175
Dividing both sides by 5:
y = 35
Substituting this value into Equation 1:
x + 35 = 85
x = 85 - 35
x = 50
Therefore, the school sold 50 student tickets and 35 adult tickets.
To check our answers, we can substitute these values back into the original equations:
Equation 1: 50 + 35 = 85 (True)
Equation 2: 5(50) + 10(35) = 600 (True)
The answers are correct.
14. The grocery store sells small bags of apples for $5 and large bags of apples for $8. You bought 6 bags and spent $45. We can set up the following system of equations:
Equation 1: x + y = 6 (Total number of bags purchased)
Equation 2: 5x + 8y = 45 (Total amount spent)
Again, let's use the method of substitution:
From Equation 1, we can rewrite it as x = 6 - y.
Substituting x = 6 - y into Equation 2:
5(6 - y) + 8y = 45
Simplifying the equation:
30 - 5y + 8y = 45
Combining like terms:
3y + 30 = 45
Subtracting 30 from both sides:
3y = 45 - 30
3y = 15
Dividing both sides by 3:
y = 5
Substituting this value into Equation 1:
x + 5 = 6
x = 6 - 5
x = 1
Therefore, you bought 1 small bag and 5 large bags of apples.
To check our answers, we can substitute these values back into the original equations:
Equation 1: 1 + 5 = 6 (True)
Equation 2: 5(1) + 8(5) = 45 (True)
The answers are correct.