Question

a prepaid cell phone charges a preset number of minutes to use text messaging. the graph represents y, the number of minutes used for x, the number of text messages sent and received. is there a direct variation? which equation represents the relationship? options: yes, y=2x. yes, y=20x. no, y=x+10. no, y=x+20.

Explanation:

Step1: Identify the points

From the graph, the points are (20,10), (40,20), (60,30), (80,40).

Step2: Check direct variation

Direct variation is \( y = kx \), so \( k=\frac{y}{x} \).
For (20,10): \( k=\frac{10}{20}=0.5 \)
For (40,20): \( k=\frac{20}{40}=0.5 \)
For (60,30): \( k=\frac{30}{60}=0.5 \)
For (80,40): \( k=\frac{40}{80}=0.5 \)
Wait, but the options have \( y = 2x \)? Wait no, maybe I mixed x and y. Wait the x-axis is Number of Text Messages, y-axis is Number of Minutes Used? Wait the labels: x is Number of Text Messages, y is Number of Minutes Used? Wait the problem says: "x, the number of minutes used for, y the number of text messages sent and received"? Wait no, re-reading: "x, the number of minutes used for, y the number of text messages sent and received". Wait the graph: x-axis (horizontal) is Number of Minutes Used (y-axis label? Wait no, the vertical axis is Number of Text Messages, horizontal is Number of Minutes Used? Wait the graph has x as Number of Text Messages (vertical) and y as Number of Minutes Used (horizontal)? Wait no, the labels: left vertical is Number of Text Messages (x-axis?), bottom horizontal is x, top horizontal is y (Number of Minutes Used). Wait the points: when x (Number of Text Messages) is 10, y (Minutes) is 20? Wait no, the red dots: first dot at (20,10) – wait, maybe x is Number of Minutes Used (horizontal axis, y-axis label), y is Number of Text Messages (vertical axis, x-axis label)? No, the problem says: "x, the number of minutes used for, y the number of text messages sent and received". So x is minutes, y is text messages. So the points are (x,y) where x (minutes) is 20, y (texts) is 10; x=40, y=20; x=60, y=30; x=80, y=40. So \( y=\frac{1}{2}x \) or \( x = 2y \), but the options: check the options. Wait the options are: Yes, y=2x; Yes, y=20x; No, y=x+10; No, y=x+20. Wait maybe I got x and y reversed. Let's re-express: if x is text messages, y is minutes. So points: (10,20), (20,40), (30,60), (40,80). Then \( y = 2x \). Let's check: when x=10, y=20 (210=20); x=20, y=40 (220=40); x=30, y=60 (230=60); x=40, y=80 (240=80). Yes! So that's direct variation with \( y = 2x \).

Answer:

Yes, \( y = 2x \) (the first option: Yes, y = 2x)