91Ó°ÊÓ

Given the point \((x_1, y_1) = (4, 5)\) and slope \(m = -\frac{2}{3}\), we can use the point-slope form of the equation of the line: \(y - y_1 = m(x - x_1)\) Substitute the given point and slope into the equation: \(y - 5 = -\frac{2}{3}(x - 4)\)

Now, we will convert the equation into slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. First, distribute -\(m\) across \((x - x_1)\): \(y - 5 = -\frac{2}{3}x + \frac{8}{3}\) Next, add 5 to both sides to isolate y: \(y = -\frac{2}{3}x + \frac{8}{3} + 5\) Since 5 can be written as the fraction \(\frac{15}{3}\), we can combine the fractions: \(y = -\frac{2}{3}x + \frac{23}{3}\) Now, we have the slope-intercept form of the equation of the line: \(y = -\frac{2}{3}x + \frac{23}{3}\)

With the equation of the line in slope-intercept form, we can easily plot the line. 1. Start by plotting the y-intercept, which is \(\frac{23}{3}\) or approximately 7.67. This point will be \((0, \frac{23}{3})\) 2. From this y-intercept, use the slope to find another point on the line. The slope is -\(\frac{2}{3}\), which means that for every 3 units to the right (in the positive x direction), the line goes down 2 units. 3. Given that, move 3 units to the right and 2 units down from the y-intercept to find another point on the line: (3, \(\frac{23}{3} - 2\)) or (3, \(\frac{17}{3}\)). 4. Connect the points (0, \(\frac{23}{3}\)) and (3, \(\frac{17}{3}\)) with a straight line, which is the graph of the line containing the point (4, 5) and with the given slope -\(\frac{2}{3}\). Now, you have the graph of the line with the given point (4, 5) and slope \(m = -\frac{2}{3}\).