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 number of text messages sent and received. is there a direct variatio equation represents the relationship? no, ( y = x + 10 ). no, ( y = x + 20 ). yes, ( y = 20x ). yes, ( y = 2x ).

Explanation:

Step1: Recall the form of direct variation

The equation for direct variation is \(y = kx\) (where \(k\) is the constant of variation), and its graph passes through the origin \((0,0)\).

Step2: Check the points on the graph

Let's take two points from the graph. Assume the points are \((10,20)\) and \((20,40)\) and \((30,60)\) and \((40,80)\).
For the point \((10,20)\), if \(y = kx\), then \(k=\frac{y}{x}=\frac{20}{10} = 2\).
For the point \((20,40)\), \(k=\frac{y}{x}=\frac{40}{20}=2\).
For the point \((30,60)\), \(k = \frac{y}{x}=\frac{60}{30}=2\).
For the point \((40,80)\), \(k=\frac{y}{x}=\frac{80}{40}=2\).
The equation \(y = 2x\) satisfies all the points on the graph and is of the form \(y=kx\) (direct - variation form).

Answer:

Yes, \(y = 2x\).