Emil and Nathaniel went into a store to buy spinners and bouncy balls. Emil bought 2 spinners and 3 bouncy balls for $4 and Nathaniel bought 1 spinner and 4 bouncy balls for $3. 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.
Let's assume the price of one spinner is x and the price of one bouncy ball is y.
According to the given information, we can create two equations:
2x + 3y = 4 - Equation 1 (Emil's purchase)
x + 4y = 3 - Equation 2 (Nathaniel's purchase)
To solve this system of equations algebraically, we can use the substitution method.
From Equation 2, we can express x in terms of y:
x = 3 - 4y
Now, substitute x in Equation 1 with this expression:
2(3 - 4y) + 3y = 4
Simplify:
6 - 8y + 3y = 4
Combine like terms:
-5y + 6 = 4
Subtract 6 from both sides:
-5y = -2
Divide both sides by -5:
y = 2/5
Now, substitute this value of y back into Equation 2 to find x:
x + 4(2/5) = 3
Multiply 4 and (2/5):
x + 8/5 = 3
Subtract 8/5 from both sides:
x = 3 - 8/5
Convert 3 to 15/5:
x = 15/5 - 8/5
Simplify:
x = 7/5
Therefore, the price of one spinner is 7/5 (or $1.40) and the price of one bouncy ball is 2/5 (or $0.40).