To determine the equation of the straight line in the given graph, we need to identify two key elements:

1. The slope (m): This can be calculated by identifying two distinct points on the line and using the formula: $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of two points on the line.

2. The y-intercept (c): This is the point where the line crosses the y-axis. It's the value of $y$ when $x = 0$.

Let's examine the graph carefully to find these details and write the equation in the form $y = mx + c$.

I'll inspect the image and calculate the equation. PIL import Image

Load the image

img_path = '/mnt/data/1740952059742usnww19r.jpg' img = Image.open(img_path)

Display the image to examine the graph

img.show()I have displayed the image for you to inspect the graph. Now, let me calculate the slope and the y-intercept to determine the equation of the line.

From the graph, I will identify two points to calculate the slope and then find the y-intercept. I'll summarize the results in a moment.Upon inspecting the graph, I identified two points on the line:

• Point 1: $(-3, 3)$
• Point 2: $(2, -2)$

Now, let's calculate the slope $m$ using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the values from the points:

$m = \frac{-2 - 3}{2 - (-3)} = \frac{-5}{5} = -1$

Next, we find the y-intercept $c$. Using the slope-intercept form $y = mx + c$, we can substitute one of the points (say, $(-3, 3)$) to find $c$: $3 = (-1)(-3) + c \quad \Rightarrow \quad 3 = 3 + c \quad \Rightarrow \quad c = 0$

Thus, the equation of the line is: $y = -x$

Would you like further details on the calculation process or have any other questions? Here are some related questions you may find helpful:

1. How do you find the slope of a line if only one point is given?
2. What happens to the equation if the line is horizontal or vertical?
3. How can you use the slope to predict the value of $y$ for a given $x$?
4. What is the significance of the y-intercept in real-world applications?
5. How can you graph a line given its equation in slope-intercept form?

Tip: Always ensure that you select clear points on the graph when calculating the slope, as small errors in choosing points can lead to inaccurate results.