Eula needs to buy binders that cost $4 each and notebooks that cost $5 each. She has $520. The inequality 4r + 5y ≤ 520 represents the situation, where r represents the number of binders and y represents the number of notebooks. Step 1: Identify the x and y-intercepts of the boundary line. When r = 0, y = 104. When y = 0, r = 130. Check: None
Eula needs to buy binders that cost
5 each. She has $520. The inequality 4r + 5y ≤ 520 represents the situation, where r represents the number of binders and y represents the number of notebooks. Step 1: Identify the x and y-intercepts of the boundary line. When r = 0, y = 104. When y = 0, r = 130. Check: None
The Correct Answer and Explanation is:
Step 1: Identify the x- and y-intercepts of the boundary line.
The inequality given is: 4r+5y≤5204r + 5y \leq 520
This inequality represents Eula’s total spending. The boundary line is the equation: 4r+5y=5204r + 5y = 520
To find the intercepts:
1. y-intercept (when r = 0):
Substitute r=0r = 0 into the equation: 4(0)+5y=520⇒5y=520⇒y=5205=1044(0) + 5y = 520 \Rightarrow 5y = 520 \Rightarrow y = \frac{520}{5} = 104
So, the y-intercept is (0,104)(0, 104).
2. x-intercept (when y = 0):
Substitute y=0y = 0 into the equation: 4r+5(0)=520⇒4r=520⇒r=5204=1304r + 5(0) = 520 \Rightarrow 4r = 520 \Rightarrow r = \frac{520}{4} = 130
So, the x-intercept is (130,0)(130, 0).
✅ These values are correct:
- y-intercept: (0, 104)
- x-intercept: (130, 0)
✅ Explanation
This problem models a real-world budgeting situation using a linear inequality. Eula is deciding how many binders and notebooks to buy without exceeding a total budget of $520. Each binder costs $4 (represented by variable rr), and each notebook costs $5 (represented by variable yy). The inequality: 4r+5y≤5204r + 5y \leq 520
represents all the combinations of binders and notebooks that she can afford.
The boundary line 4r+5y=5204r + 5y = 520 divides the coordinate plane into two regions: one where the total cost is less than or equal to $520 (the feasible region), and one where it is more (which is not allowed). The intercepts help us graph this line and understand the limits.
The x-intercept tells us how many binders Eula can buy if she buys no notebooks—this is 130 binders. The y-intercept tells us how many notebooks she can buy if she buys no binders—this is 104 notebooks. Any point (r, y) that lies on or below the line represents a valid combination of binders and notebooks she can purchase.
Understanding these intercepts is crucial for visualizing the budget constraint. They define the edge of what’s possible, and the area beneath the line includes all affordable combinations. By using intercepts, Eula can make informed decisions, whether she prefers more binders, more notebooks, or a mix of both—while staying within budget.
