To analyze the given image, I will extract the mathematical details of the line shown on the coordinate grid. Let's identify the line's slope and y-intercept, and derive its equation.

Steps:

1. Determine the slope ($m$): The slope is calculated as: $m = \frac{\Delta y}{\Delta x}$ Choose two points visible on the line to compute the slope.

2. Identify the y-intercept ($b$): The y-intercept is where the line crosses the y-axis.

3. Write the equation of the line: The equation of a straight line is given by: $y = mx + b$

Let me analyze the image further to derive the equation.From the provided image, here are the key observations and steps to determine the equation of the line:

1. Identifying Two Points on the Line

The line passes through clear points on the grid:

• Point 1: $(-4, -4)$
• Point 2: $(0, 4)$

2. Calculating the Slope

The slope ($m$) is calculated as: $m = \frac{\Delta y}{\Delta x} = \frac{4 - (-4)}{0 - (-4)} = \frac{8}{4} = 2$

3. Identifying the y-Intercept ($b$)

The line crosses the y-axis at $y = 4$, so the y-intercept is: $b = 4$

4. Equation of the Line

Using the slope-intercept form $y = mx + b$, the equation of the line is: $y = 2x + 4$

Summary

The equation of the line is: $\boxed{y = 2x + 4}$

Would you like further explanations or details on any of the steps?

Related Questions

1. How do you calculate the slope if the points are not clearly grid-aligned?
2. What happens if the line is vertical? What would its equation look like?
3. How can this slope-intercept equation be rewritten in standard form?
4. How do you find the x-intercept of a line given its equation?
5. What does it mean if two lines have the same slope but different y-intercepts?

Tip:

When determining the slope of a line, always ensure that you simplify the fraction representing $\Delta y / \Delta x$ for accurate results.