Question

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

Explanation:

Step1: Recall the form of direct variation

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

Step2: Check if the graph passes through the origin

Looking at the graph, when \(x = 0\) (number of text messages is \(0\)), the value of \(y\) (number of minutes) is not \(0\). So, it is not a direct variation.

Step3: Find the equation of the line

Let's assume two points from the graph. Suppose the points are \((10,30)\) and \((20,40)\).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{40 - 30}{20 - 10}=1\).
Using the point - slope form \(y - y_1=m(x - x_1)\), with \(m = 1\) and \((x_1,y_1)=(10,30)\)
\(y-30=1\times(x - 10)\)
\(y=x+20\)

Answer:

No, \(y = x + 20\)