Emil and Nathaniel went into a store to buy spinners and bouncy balls. Emil bought 2 spinners and 3 bouncy balls for $4.20 and Nathaniel bought 1 spinner and 4 bouncy balls for $3.60. Write a system of equations to represent this situation. Solve the system of equations algebraically to determine the price of each spinner and the price of each bouncy ball. Explain how you chose which algebraic strategy – substitution or elimination – to use to solve this system.
Let x be the price of a spinner and y be the price of a bouncy ball.
From the information given, we can create two equations:
2x + 3y = 4.20 (equation 1) <- Emil's purchase
x + 4y = 3.60 (equation 2) <- Nathaniel's purchase
To solve this system of equations, we can use either substitution or elimination method.
Since the coefficients of x in equation 1 and equation 2 are both 1, it will be easier to use the elimination method. We can multiply equation 2 by 2 to make the coefficients of x in both equations the same:
2(x + 4y) = 2(3.60)
2x + 8y = 7.20 (equation 3)
Now we can subtract equation 1 from equation 3 to eliminate x:
(2x + 8y) - (2x + 3y) = 7.20 - 4.20
5y = 3
Therefore, we can conclude that the price of a bouncy ball, y, is $0.60.
Substituting this value into equation 2:
x + 4(0.60) = 3.60
x + 2.40 = 3.60
x = 3.60 - 2.40
x = 1.20
Therefore, the price of a spinner, x, is $1.20.
So, each spinner costs $1.20 and each bouncy ball costs $0.60.
I chose to use the elimination method because the coefficients of x in both equations were already 1, which made it easier to eliminate x when subtracting the equations.