Question

find the slope of the line.
graph of a line on a coordinate plane with grid, and multiple - choice options: m = 2, $m = -\frac{1}{2}$, m = -1, $m=\frac{-3}{4}$

Explanation:

Step1: Identify two points on the line

Let's take two clear points on the blue line. From the graph, we can see two points: let's say \((-4, 1)\) and \((-2, 0)\) (or we can also use other visible points, but these are easy to identify). Wait, actually, looking at the grid, let's find two points with integer coordinates. Let's pick the first blue dot: let's assume its coordinates are \((-4, 1)\) and the second one (the lower blue dot) is \((-2, 0)\)? Wait, no, maybe better to take two points. Wait, another way: the slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's find two points. Let's take the upper blue point: let's say \((-4, 1)\) and the lower blue point: \((-2, 0)\)? Wait, no, maybe I made a mistake. Wait, looking at the line, let's take two points. Let's see, when x increases by 2, y decreases by 1? Wait, no, let's check the grid. Let's take two points: let's say \((-4, 1)\) and \((-2, 0)\). Wait, the difference in y: \(0 - 1 = -1\), difference in x: \(-2 - (-4) = 2\). So slope \(m=\frac{-1}{2}=-\frac{1}{2}\). Wait, but let's confirm with another pair. Let's take \((-2, 0)\) and \((0, -1)\). Then \(y_2 - y_1 = -1 - 0 = -1\), \(x_2 - x_1 = 0 - (-2) = 2\), so slope is \(\frac{-1}{2}=-\frac{1}{2}\). So that's the slope.

Step2: Calculate the slope using the formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points on the line, say \((x_1, y_1)=(-4, 1)\) and \((x_2, y_2)=(-2, 0)\). Then \(y_2 - y_1 = 0 - 1 = -1\) and \(x_2 - x_1 = -2 - (-4) = 2\). So \(m=\frac{-1}{2}=-\frac{1}{2}\).

Answer:

\(m = -\frac{1}{2}\) (corresponding to the option \(m = -\frac{1}{2}\))