Explanation:
We are given two points from the problem:
- For 8 people, cost = \$123 → point: (8, 123)
- For 12 people, cost = \$167 → point: (12, 167)
We want a linear equation in slope-intercept form:
y = mx + b, where:
- y = total cost
- x = number of people
- m = slope (rate of change, i.e., cost per person)
- b = y-intercept (fixed cost when x = 0 — maybe setup fee, etc.)
---
Step 1: Find the slope (m)
Slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{167 - 123}{12 - 8} = \frac{44}{4} = 11
\]
So the slope is
11. That means the cost increases by \$11 for each additional person.
---
Step 2: Find the y-intercept (b)
Use one point and plug into y = mx + b. Let’s use (8, 123):
\[
123 = 11(8) + b \\
123 = 88 + b \\
b = 123 - 88 = 35
\]
So the y-intercept is
35.
That means even with 0 people, there’s a \$35 base cost (maybe rental fee or setup).
---
Step 3: Write the equation
\[
y = 11x + 35
\]
---
Step 4: Interpret the slope
The slope is
11, which represents
\$11 per person — the cost increases by \$11 for each extra person.
---
Step 5: Cost for 26 people
Plug x = 26 into the equation:
\[
y = 11(26) + 35 = 286 + 35 = 321
\]
So a party for 26 people costs
\$321.
Final Answer:
Equation in slope-intercept form:
y = 11x + 35
The slope is
11, which represents
\$11 per person
A party for 26 people will cost
\$321
Parent Tip: Review the logic above to help your child master the concept of slope intercept form word problems worksheet.
