Let's analyze the given equation [tex]\(\hat{y} = 12x + 30\)[/tex] to identify and interpret the slope in the context of the bowling league costs.
### Step-by-Step Solution:
1. Identifying the components of the equation:
- The given equation is in the slope-intercept form, [tex]\(\hat{y} = mx + b\)[/tex], where:
- [tex]\(m\)[/tex] is the slope.
- [tex]\(b\)[/tex] is the y-intercept.
2. Finding the slope [tex]\(m\)[/tex]:
- In the equation [tex]\(\hat{y} = 12x + 30\)[/tex], the coefficient of [tex]\(x\)[/tex] is 12.
- Thus, the slope [tex]\(m\)[/tex] is 12.
3. Interpreting the y-intercept [tex]\(b\)[/tex]:
- The constant term is 30.
- Thus, the y-intercept [tex]\(b\)[/tex] is 30. This represents the one-time registration fee.
4. Interpreting the slope [tex]\(m\)[/tex]:
- The slope [tex]\(m = 12\)[/tex] represents the rate of change of the cost with respect to the number of months.
- This means that for each month, the cost increases by [tex]$12.
5. Combining the interpretations:
- Starting with a one-time registration fee of $[/tex]30, the monthly cost increases by [tex]$12.
### Conclusion:
The slope is 12. Starting at $[/tex]30, the cost will increase by $12 per month.
