An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Set up the system of equations: x + 2 y = 82 and 3 x + y = 96 .
Solve for x by eliminating y : x = 22 .
Substitute x into one of the equations to solve for y : y = 30 .
Calculate the difference y − x to find the excess price: 8 .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y , representing the prices of silver and gold tickets, respectively. Our goal is to find the difference between the price of a gold ticket and a silver ticket, which is y − x .
Setting up the Equations The given system of equations is:
x + 2 y = 82
3 x + y = 96
We can solve this system using the substitution or elimination method. Let's use the elimination method. Multiply the second equation by 2 to eliminate y :
2 ( 3 x + y ) = 2 ( 96 )
6 x + 2 y = 192
Eliminating y Now we have the following system:
x + 2 y = 82
6 x + 2 y = 192
Subtract the first equation from the second equation to eliminate y :
( 6 x + 2 y ) − ( x + 2 y ) = 192 − 82
5 x = 110
Solving for x Divide by 5 to solve for x :
x = 5 110
x = 22
Substituting x to find y Now that we have the value of x , we can substitute it into either of the original equations to solve for y . Let's use the first equation:
x + 2 y = 82
22 + 2 y = 82
Isolating y Subtract 22 from both sides:
2 y = 82 − 22
2 y = 60
Solving for y Divide by 2 to solve for y :
y = 2 60
y = 30
Finding the Difference Now we have the values of x and y : x = 22 and y = 30 . We want to find the difference y − x :
y − x = 30 − 22
y − x = 8
Final Answer The price of a ticket for the gold section exceeds the price for a ticket for the silver section by $8 .
Examples
Understanding systems of equations can help in various real-life scenarios, such as determining the cost of different items when given combined prices. For instance, if you know the total cost of a certain number of apples and bananas and you have another set of combined quantities and their total cost, you can use a system of equations to find the individual prices of apples and bananas. This method is also applicable in business for cost analysis, resource allocation, and determining optimal pricing strategies.
