Let’s work through this step by step.
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Step 1: Understand the problem
We need to seat 180 people using two types of tables:
- Small tables seat 4 people each.
- Large tables seat 6 people each.
We don’t know how many of each table we’ll use — let’s call:
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x = number of small tables
-
y = number of large tables
So, total seats = 4x + 6y
And we need that to equal 180.
→ So the equation is:
4x + 6y = 180
We can simplify it by dividing everything by 2:
2x + 3y = 90
(This is easier to graph and work with, but both are correct. We’ll use the simplified version for intercepts and graphing.)
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Step 2: Find the intercepts
The x-intercept is where y = 0.
Plug into 2x + 3y = 90 → 2x + 0 = 90 → x = 45
→ x-intercept:
(45, 0)
The y-intercept is where x = 0.
Plug into 2x + 3y = 90 → 0 + 3y = 90 → y = 30
→ y-intercept:
(0, 30)
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Step 3: Graph the equation
We plot the line using the two intercepts: (45, 0) and (0, 30).
Since we’re dealing with real-world items (tables), we only care about the part of the line in the first quadrant (positive x and y values), and only whole numbers (you can’t rent half a table).
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Step 4: Give 3 possible combinations of tables rented
We need integer solutions to 2x + 3y = 90.
Let’s pick some values:
- If x = 0 → 3y = 90 → y = 30 → (0 small, 30 large)
- If x = 15 → 2(15) = 30 → 3y = 60 → y = 20 → (15 small, 20 large)
- If x = 30 → 2(30) = 60 → 3y = 30 → y = 10 → (30 small, 10 large)
These all work!
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Step 5: State the domain and range
Domain = all possible x-values (number of small tables) that make sense.
Since x must be a non-negative integer, and from the equation:
When y = 0 → x = 45
When x = 0 → y = 30
Also, since 2x + 3y = 90, x must be such that (90 - 2x) is divisible by 3 and ≥ 0.
But for simplicity in this context (and since the graph is continuous even though reality is discrete), we usually state the domain as the interval between the intercepts on the x-axis.
→ Domain:
0 ≤ x ≤ 45
Range: all possible y-values (number of large tables)
→ Range:
0 ≤ y ≤ 30
*(Note: In reality, x and y must be integers, but unless specified, we often give the continuous intervals for domain/range in graphing problems like this.)*
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Step 6: Explain what the intercepts mean
-
x-intercept (45, 0): This means if you use 45 small tables and 0 large tables, you seat exactly 180 people.
-
y-intercept (0, 30): This means if you use 0 small tables and 30 large tables, you seat exactly 180 people.
They represent the extreme cases where you use only one type of table.
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Final Answer:
Equation: 4x + 6y = 180 (or simplified: 2x + 3y = 90)
x-intercept: (45, 0)
y-intercept: (0, 30)
Three possible combinations:
- 0 small tables and 30 large tables
- 15 small tables and 20 large tables
- 30 small tables and 10 large tables
Domain: 0 ≤ x ≤ 45
Range: 0 ≤ y ≤ 30
Intercepts meaning:
- x-intercept: Using only small tables (45 of them) seats everyone.
- y-intercept: Using only large tables (30 of them) seats everyone.
Parent Tip: Review the logic above to help your child master the concept of standard form word problems worksheet.
