Question

what is the equation of the line?
options: y = 1.5x, y = 3x, y = 15x, y = 30x
chart: x - axis (number of months) from 0 to 4, y - axis (total cost ($)) from 0 to 80, line passes through (0,0) and other grid points.

Explanation:

Step1: Recall slope-intercept form

The equation of a line in slope - intercept form is \(y = mx\) (since the line passes through the origin, \(b = 0\)), where \(m\) is the slope. The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Find two points on the line

From the graph, when \(x = 1\) (1 month), let's find the corresponding \(y\) - value. Looking at the grid, when \(x = 1\), \(y = 30\)? Wait, no, wait. Wait, the \(x\) - axis is the number of months (x) and the \(y\) - axis is total cost (y). Wait, let's take two points. When \(x = 1\), what's \(y\)? Wait, maybe I misread the axes. Wait, the \(x\) - axis is "Number of Months" (so \(x\)) and \(y\) - axis is "Total Cost (\$)" (so \(y\)). Let's take \(x = 2\), what's \(y\)? Wait, looking at the line, when \(x = 2\), \(y = 30\)? Wait, no, let's check the options. The options are \(y = 1.5x\), \(y=15x\), \(y = 3x\), \(y = 30x\). Wait, maybe I made a mistake in axis interpretation. Wait, maybe the \(x\) - axis is the number of months, and when \(x = 1\), \(y = 30\)? Wait, no, let's calculate the slope. Let's take two points: \((0,0)\) and \((2,30)\). Then the slope \(m=\frac{30 - 0}{2 - 0}=\frac{30}{2}=15\)? Wait, no, \((2,30)\): if \(x = 2\) (2 months), \(y = 30\) dollars. Then \(m=\frac{y}{x}=\frac{30}{2}=15\)? Wait, no, wait the options have \(y = 15x\), \(y=30x\) etc. Wait, let's take \(x = 1\), if the line passes through \((1,30)\), then \(y = 30x\) would give \(y = 30\) when \(x = 1\), \(y = 60\) when \(x = 2\), which matches the line? Wait, no, the line in the graph: when \(x = 2\) (2 months), \(y = 30\)? Wait, maybe the axes are reversed? Wait, the problem says "What is the equation of the line?" with the graph having \(x\) as number of months and \(y\) as total cost. Wait, let's check the slope again. Let's take \(x = 1\), \(y = 30\) (from the line). Then for \(y=mx\), \(m=\frac{y}{x}=\frac{30}{1}=30\)? No, that can't be. Wait, maybe the \(x\) - axis is the number of items and \(y\) is cost, but no, the label is number of months. Wait, maybe I misread the graph. Wait, the options are \(y = 1.5x\), \(y = 15x\), \(y = 3x\), \(y = 30x\). Let's check \(y = 30x\): when \(x = 1\), \(y = 30\); \(x = 2\), \(y = 60\); \(x = 3\), \(y = 90\). But the graph's \(y\) - axis goes up to 80. Wait, maybe the \(x\) - axis is the number of months, and when \(x = 2\), \(y = 30\). Then \(m=\frac{30}{2}=15\), so \(y = 15x\). Let's check: \(x = 1\), \(y = 15\); \(x = 2\), \(y = 30\); \(x = 3\), \(y = 45\). But the graph's \(y\) - axis goes up to 80, so 45 is within. Wait, maybe the line passes through \((2,30)\), so \(y = 15x\) (since \(15\times2 = 30\)). Yes, that makes sense. So the slope \(m = 15\), so the equation is \(y = 15x\).

Answer:

\(y = 15x\)