Question

the slope is select select 3/2 2/3 3 2

Explanation:

Step1: Identify two points on the line

Let's assume the left - most point is \((x_1,y_1)=(2,2)\) and the right - most point is \((x_2,y_2)=(8,6)\) (by counting the grid squares. The horizontal distance between the two points: \(x_2 - x_1=8 - 2 = 6\), the vertical distance: \(y_2 - y_1=6 - 2 = 4\)? Wait, no, maybe a better way. Let's take two clear points. Let's say the starting point (left) is at \((2,2)\) and the ending point (right) is at \((8,6)\)? Wait, no, let's count the rise over run. Let's find two points where the line passes through the intersection of the grid lines. Suppose the left point is \((2,2)\) and the right point is \((8,6)\)? Wait, no, let's do it properly. The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's pick two points: let's say the left - most point is \((2,2)\) and the right - most point is \((8,6)\)? Wait, no, maybe the left point is \((2,2)\) and the right point is \((5,4)\)? Wait, no, let's count the number of units we move up (rise) and the number of units we move right (run). Let's take the left point as \((2,2)\) and the right point as \((8,6)\). Then \(y_2 - y_1=6 - 2 = 4\), \(x_2 - x_1=8 - 2 = 6\), so \(m = \frac{4}{6}=\frac{2}{3}\)? Wait, no, maybe I made a mistake. Wait, let's take another pair. Let's say the left point is \((2,2)\) and the right point is \((5,4)\). Then \(y_2 - y_1 = 4 - 2=2\), \(x_2 - x_1=5 - 2 = 3\), so \(m=\frac{2}{3}\). Wait, or maybe the left point is \((2,2)\) and the right point is \((8,6)\), then \(y_2 - y_1=4\), \(x_2 - x_1 = 6\), \(\frac{4}{6}=\frac{2}{3}\). Wait, but let's check again. Let's take the left point as \((2,2)\) and the right point as \((8,6)\): rise is \(6 - 2 = 4\), run is \(8 - 2 = 6\), so slope is \(\frac{4}{6}=\frac{2}{3}\). Alternatively, if we take the left point as \((2,2)\) and the right point as \((5,4)\): rise \(4 - 2 = 2\), run \(5 - 2 = 3\), slope \(\frac{2}{3}\). So the slope is \(\frac{2}{3}\). Wait, but let's check the options. The options are \(3/2\), \(2/3\), \(3\), \(2\). So the correct slope is \(2/3\).

Wait, maybe I messed up the points. Let's take two points: let's say the left point is \((2,2)\) and the right point is \((8,6)\). Then \(y_2 - y_1=6 - 2 = 4\), \(x_2 - x_1=8 - 2 = 6\), so \(m=\frac{4}{6}=\frac{2}{3}\). Yes, that's correct. So the slope is \(2/3\).

Step1: Recall the slope formula

The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Identify two points on the line

Let's choose two points on the line that lie on the grid intersections. Let the left - most point be \((x_1,y_1)=(2,2)\) and the right - most point be \((x_2,y_2)=(8,6)\) (we can also choose other pairs, but this is a valid pair).

Step3: Calculate the rise and run

The rise (change in \(y\)) is \(y_2 - y_1=6 - 2 = 4\). The run (change in \(x\)) is \(x_2 - x_1=8 - 2 = 6\).

Step4: Calculate the slope

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we substitute the values: \(m=\frac{4}{6}=\frac{2}{3}\).

Answer:

\(\frac{2}{3}\) (corresponding to the option "2/3")