Question

graph the line with y-intercept 3 and slope \\(\frac{2}{3}\\).

Explanation:

Step1: Identify y - intercept point

The y - intercept is 3, so the point on the y - axis is \((0,3)\). Plot this point on the coordinate plane.

Step2: Use slope to find next point

The slope \(m=\frac{2}{3}\), which means for a run (change in \(x\)) of 3 units, the rise (change in \(y\)) is 2 units. Starting from \((0,3)\), move 3 units to the right (increase \(x\) by 3) and 2 units up (increase \(y\) by 2). So the new point is \((0 + 3,3+2)=(3,5)\). We can also move in the opposite direction: from \((0,3)\), move 3 units to the left (decrease \(x\) by 3) and 2 units down (decrease \(y\) by 2) to get \((- 3,1)\).

Step3: Draw the line

Connect the points \((0,3)\), \((3,5)\), \((-3,1)\) (and other points found using the slope) with a straight line to graph the line \(y=\frac{2}{3}x + 3\).

Answer:

The line is graphed by first plotting \((0,3)\), then using the slope \(\frac{2}{3}\) to find additional points (e.g., \((3,5)\), \((-3,1)\)) and drawing a straight line through them.