Question: User: 2. Bowling
Bowling Alley A charges a $25 lane fee plus $4 per game. Bowling Alley B charges a $10 lane fee plus $7 per game.
Write a system of equations to represent this relationship. Graph the equations and use the graph to determine the number of games that would cost the same at each bowling alley.
Be sure to use let statements to define variables!
1. **State the problem:**
We want to find the number of games where the total cost at Bowling Alley A equals the total cost at Bowling Alley B.
2. **Define variables:**
Let $x$ = number of games played.
Let $y$ = total cost.
3. **Write the system of equations:**
For Bowling Alley A: $$y = 25 + 4x$$
For Bowling Alley B: $$y = 10 + 7x$$
4. **Set the equations equal to find when costs are the same:**
$$25 + 4x = 10 + 7x$$
5. **Solve for $x$:**
Subtract $10$ from both sides:
$$25 + 4x - 10 = 7x$$
$$15 + 4x = 7x$$
Subtract $4x$ from both sides:
$$15 + \cancel{4x} - \cancel{4x} = 7x - 4x$$
$$15 = 3x$$
Divide both sides by $3$:
$$\frac{15}{\cancel{3}} = \frac{3x}{\cancel{3}}$$
$$5 = x$$
6. **Interpretation:**
The cost is the same when $x = 5$ games are played.
7. **Find the cost at $x=5$:**
Substitute $x=5$ into either equation:
$$y = 25 + 4(5) = 25 + 20 = 45$$
So, the cost is $45$ for 5 games at both alleys.
8. **Graphing:**
The graph of the two lines $y=25+4x$ and $y=10+7x$ intersect at the point $(5, 45)$, confirming our solution.
**Final answer:**
The number of games that cost the same at both alleys is $$\boxed{5}$$ games, costing $$\boxed{45}$$.