To graph the line that passes through the points $(0, 7)$ and $(5, 3)$, we will first determine the equation of the line in slope-intercept form $y = mx + b$, where:

• $m$ is the slope of the line,
• $b$ is the y-intercept.

Step 1: Calculate the slope $m$

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For the points $(0, 7)$ and $(5, 3)$, we substitute the coordinates into the formula:

$m = \frac{3 - 7}{5 - 0} = \frac{-4}{5}$

So, the slope of the line is $m = -\frac{4}{5}$.

Step 2: Determine the y-intercept $b$

Now, we use the slope $m = -\frac{4}{5}$ and one of the points (let’s use $(0, 7)$) to find the y-intercept. Since the point $(0, 7)$ is on the y-axis, the y-intercept is simply the $y$-coordinate of this point.

Thus, the y-intercept $b = 7$.

Step 3: Write the equation of the line

Now that we have the slope and y-intercept, we can write the equation of the line:

$y = -\frac{4}{5}x + 7$

Step 4: Graph the line

• Start by plotting the y-intercept at $(0, 7)$.
• Then use the slope $-\frac{4}{5}$, meaning for every 5 units you move to the right, you move 4 units down (since the slope is negative).
• Another point on the line can be found by moving from $(0, 7)$ five units to the right (to $x = 5$), and then moving four units down to $y = 3$, which is the second point $(5, 3)$.

These two points $(0, 7)$ and $(5, 3)$ are enough to graph the line.

Final Answer:

• The equation of the line is $y = -\frac{4}{5}x + 7$.
• You can plot the points $(0, 7)$ and $(5, 3)$, then draw a straight line through them to visualize the graph.

Would you like me to go further with any of the steps, or do you have another question? Here are 5 related questions to help expand the concept:

1. How do you find the equation of a line if the slope and one point are given?
2. What does the slope of a line represent geometrically?
3. How do you graph a line if you're given its equation in standard form $Ax + By = C$?
4. What is the point-slope form of a line, and how can it be used to write an equation?
5. How can you calculate the slope of a line parallel or perpendicular to the given line?

Tip: When graphing a line, always plot at least two points, especially if one of the points is the y-intercept. This gives you a clear reference for drawing the line accurately!