Question

graph the line with the equation $y = \frac{2}{3}x + 4$.

Explanation:

Step1: Identify the y - intercept

The equation of the line is in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the equation \(y=\frac{2}{3}x + 4\), the y - intercept \(b = 4\). This means the line crosses the y - axis at the point \((0,4)\). We can plot this point on the graph.

Step2: Determine the slope

The slope \(m=\frac{2}{3}\). The slope is defined as \(\frac{\text{rise}}{\text{run}}\), which means for every 3 units we move to the right (run) along the x - axis, we move up 2 units (rise) along the y - axis. Starting from the y - intercept \((0,4)\), if we move 3 units to the right (to \(x = 3\)) and 2 units up, we reach the point \((3,4 + 2)=(3,6)\). We can also move in the opposite direction: from \((0,4)\), move 3 units to the left (to \(x=- 3\)) and 2 units down, we reach the point \((-3,4-2)=(-3,2)\).

Step3: Draw the line

After plotting two or more points (such as \((0,4)\), \((3,6)\), and \((-3,2)\)), we can draw a straight line passing through these points to represent the equation \(y = \frac{2}{3}x+4\).

(Note: Since the problem is about graphing, the final answer is the graph with the line passing through points like \((0,4)\), \((3,6)\), \((-3,2)\) etc. But in text - based form, we describe the process of graphing as above.)

Answer:

The line is graphed by first plotting the y - intercept \((0,4)\), then using the slope \(\frac{2}{3}\) to find additional points (e.g., \((3,6)\), \((-3,2)\)) and drawing a straight line through these points.