To sketch the graph of the linear equation $y = -\frac{2}{3}x + 5$, follow these steps:

1. Identify the slope and y-intercept:

   • The slope ($m$) is $-\frac{2}{3}$.
   • The y-intercept ($b$) is 5.

2. Plot the y-intercept:

   • Start at the point $(0, 5)$ on the y-axis.

3. Use the slope to find another point:

   • The slope $-\frac{2}{3}$ means that for every 3 units you move to the right (positive direction on the x-axis), you move 2 units down (negative direction on the y-axis).
   • From the y-intercept $(0, 5)$, move 3 units to the right to $x = 3$, and 2 units down to $y = 3$. This gives you the point $(3, 3)$.

4. Plot this second point:

   • Plot the point $(3, 3)$ on the graph.

5. Draw the line:

   • Draw a straight line through the points $(0, 5)$ and $(3, 3)$.

Here's a sketch of the graph based on these steps:

This is how you graph the linear equation $y = -\frac{2}{3}x + 5$.

Would you like more details or have any questions?

Questions to Expand Understanding:

1. How do you determine the slope from the equation of a line?
2. What is the significance of the y-intercept in a linear equation?
3. How can you graph a line if the equation is in a different form, such as $Ax + By = C$?
4. What happens to the graph if the slope is positive instead of negative?
5. How do you find the x-intercept of the equation $y = -\frac{2}{3}x + 5$?
6. How does changing the y-intercept affect the graph of the line?
7. How can you verify that a point lies on the line represented by the equation $y = -\frac{2}{3}x + 5$?
8. How would the graph change if the slope was $-1$ instead of $-\frac{2}{3}$?

Tip:

When plotting points based on the slope, remember that the direction matters: moving right and up/down corresponds to positive and negative slopes, respectively.