Sure, let's tackle this problem by setting up a system of equations and solving it step-by-step.
Step 1: Define the variables
- Let [tex]\( x \)[/tex] be the number of bowling games played.
- Let [tex]\( C \)[/tex] be the total cost of bowling.
Step 2: Write the cost equations for both bowling alleys
For Rock and Bowl, the total cost [tex]\( C \)[/tex] can be expressed as:
[tex]\[ C = 2.75x + 3.00 \][/tex]
where [tex]$2.75 is the cost per game, and $[/tex]3.00 is the shoe rental fee.
For Super Bowling, the total cost [tex]\( C \)[/tex] can be expressed as:
[tex]\[ C = 2.25x + 3.50 \][/tex]
where [tex]$2.25 is the cost per game, and $[/tex]3.50 is the shoe rental fee.
Step 3: Set the two cost equations equal to each other
To find the number of games [tex]\( x \)[/tex] where the cost is the same at both places, set the total cost equations equal to each other:
[tex]\[ 2.75x + 3.00 = 2.25x + 3.50 \][/tex]
Step 4: Solve for [tex]\( x \)[/tex]
Isolate [tex]\( x \)[/tex] by moving the terms involving [tex]\( x \)[/tex] to one side and the constant terms to the other side:
[tex]\[ 2.75x - 2.25x = 3.50 - 3.00 \][/tex]
[tex]\[ 0.50x = 0.50 \][/tex]
[tex]\[ x = 1 \][/tex]
So, after 1 game, the cost of bowling will be the same at both places.
Step 5: Calculate the total cost for one game
Substitute [tex]\( x = 1 \)[/tex] back into either of the original cost equations to find the cost.
Using Rock and Bowl's cost equation:
[tex]\[ C = 2.75(1) + 3.00 = 2.75 + 3.00 = 5.75 \][/tex]
Using Super Bowling's cost equation:
[tex]\[ C = 2.25(1) + 3.50 = 2.25 + 3.50 = 5.75 \][/tex]
Thus, the total cost after one game at both places is [tex]$5.75.
Conclusion
After 1 game, the cost of bowling will be approximately the same at both Rock and Bowl and Super Bowling, and that cost will be $[/tex]5.75.
