Question

graph the line with slope \\(\frac{2}{3}\\) passing through the point \\((-2, 3)\\).

Explanation:

Step1: Plot the given point

The given point is \((-2, 3)\). Locate \(-2\) on the x - axis and \(3\) on the y - axis, then mark the point \((-2,3)\) on the coordinate plane.

Step2: Use the slope to find another point

The slope \(m=\frac{2}{3}\), which means \(\text{rise}=\ 2\) (change in y - value) and \(\text{run}=\ 3\) (change in x - value). Starting from the point \((-2,3)\), to find the next point, we move 3 units to the right (because run is positive) along the x - axis and 2 units up (because rise is positive) along the y - axis. So the new x - coordinate is \(-2 + 3=1\) and the new y - coordinate is \(3+2 = 5\). So we get the point \((1,5)\). We can also move in the opposite direction: 3 units to the left and 2 units down. Starting from \((-2,3)\), the new x - coordinate is \(-2-3=-5\) and the new y - coordinate is \(3 - 2=1\), giving the point \((-5,1)\).

Step3: Draw the line

After plotting at least two points (e.g., \((-2,3)\) and \((1,5)\) or \((-2,3)\) and \((-5,1)\)), draw a straight line passing through these points. The line should extend in both directions to represent the infinite nature of a line.

Answer:

To graph the line:

1. Plot the point \((-2,3)\).
2. Use the slope \(\frac{2}{3}\) to find additional points (e.g., \((1,5)\) by moving 3 right and 2 up from \((-2,3)\) or \((-5,1)\) by moving 3 left and 2 down from \((-2,3)\)).
3. Draw a straight line through the plotted points, extending it in both directions.