To find the equation of the straight line shown in the image, we need to determine two things:

1. The slope (m): This is the rate at which y changes as x changes, or the "rise over run" between two points on the line.
2. The y-intercept (c): This is the point where the line crosses the y-axis.

Steps:

1. Find two points on the line:

   • Look for two clear points where the line intersects the grid.

2. Calculate the slope (m):

   • The slope formula is: $m = \frac{{\text{{change in }} y}}{{\text{{change in }} x}} = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ Using the two points you identified, plug in the values for $y_2, y_1, x_2, x_1$.

3. Find the y-intercept (c):

   • Once we know the slope (m), we can use the slope-intercept form $y = mx + c$ and plug in the coordinates of one of the points to solve for $c$.

I will now analyze the graph to find these details. matplotlib import pyplot as plt import numpy as np

Set up the image path and load it for analysis

image_path = '/mnt/data/1739125244291fa37yemm.jpg'

Display the image to manually analyze the line

img = plt.imread(image_path) plt.imshow(img) plt.axis('off') plt.show()I can observe that the straight line on the graph has two points that are easy to identify: one at $(0, 2)$ (the y-intercept) and another at $(-4, 4)$.

Step 1: Find the slope $m$

To calculate the slope $m$, use the formula: $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ Substituting the points $(x_1, y_1) = (-4, 4)$ and $(x_2, y_2) = (0, 2)$: $m = \frac{{2 - 4}}{{0 - (-4)}} = \frac{{-2}}{{4}} = -\frac{1}{2}$

Step 2: Find the y-intercept $c$

Since we know the line passes through $(0, 2)$, the y-intercept $c$ is 2.

Final Equation

Now that we have $m = -\frac{1}{2}$ and $c = 2$, the equation of the line is: $y = -\frac{1}{2}x + 2$

Would you like further clarification or details on any part of the process?

Related Questions:

1. How would you determine the slope for a horizontal or vertical line?
2. What does the slope of 0 imply about a line?
3. How can we use this equation to find points on the line for different values of $x$?
4. How would you graph an equation with a slope of $m = -\frac{1}{2}$ and a y-intercept of 2?
5. How would you calculate the slope of a line given two points in general?

Tip: When finding the equation of a line, always start by identifying the slope from two clear points and use the y-intercept directly if it's visible.