Question

graph the line that represents the equation $y = -\frac{2}{3}x + 1$.

Explanation:

Step1: Identify the y-intercept

The equation is in slope - intercept form \(y = mx + b\), where \(b\) is the y - intercept. For the equation \(y=-\frac{2}{3}x + 1\), \(b = 1\). So the line crosses the y - axis at the point \((0,1)\). We can plot this point on the graph.

Step2: Use the slope to find another point

The slope \(m=-\frac{2}{3}\). The slope is defined as \(\frac{\text{rise}}{\text{run}}\). A slope of \(-\frac{2}{3}\) means that for a run (change in \(x\)) of \(3\) units (we can choose a positive run to the right), the rise (change in \(y\)) is \(- 2\) units (downwards). Starting from the y - intercept \((0,1)\), if we move \(3\) units to the right (increase \(x\) by \(3\), so \(x=0 + 3=3\)) and \(2\) units down (decrease \(y\) by \(2\), so \(y = 1-2=-1\)), we get the point \((3,-1)\). We can also use a run of \(- 3\) (to the left) and a rise of \(2\) (upwards). Starting from \((0,1)\), moving \(3\) units to the left (\(x = 0-3=-3\)) and \(2\) units up (\(y=1 + 2 = 3\)) gives the point \((-3,3)\).

Step3: Draw the line

After plotting two points (e.g., \((0,1)\) and \((3,-1)\) or \((0,1)\) and \((-3,3)\)), we can use the 'Line' tool to draw a straight line passing through these two points.

(Note: Since this is a graph - drawing problem, the final answer is the graph of the line passing through the points found using the y - intercept and the slope. The key points to plot are \((0,1)\) and another point like \((3,-1)\) or \((-3,3)\) and then drawing the line through them.)

Answer:

The line is drawn by first plotting the point \((0,1)\) (the y - intercept) and then using the slope \(-\frac{2}{3}\) to find another point (e.g., \((3,-1)\) or \((-3,3)\)) and then drawing a straight line through these points.