To determine the equation of the line shown in the graph, let’s analyze the slope and y-intercept from the graph.

1. Identify the y-intercept: The line crosses the y-axis at $y = 1$. Therefore, the y-intercept $b$ of the line is $1$.

2. Calculate the slope (m):

   • Choose two points on the line. For instance, two visible points are $(0, 1)$ (the y-intercept) and $(7, -4)$.
   • Use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 1}{7 - 0} = \frac{-5}{7}$
   • Therefore, the slope $m$ is $-\frac{5}{7}$.

3. Formulate the equation: Using the slope-intercept form $y = mx + b$, we get: $y = -\frac{5}{7}x + 1$

This matches the first option given:

• $y = -\frac{5}{7}x + 1$

Answer

The correct answer is: $\boxed{y = -\frac{5}{7}x + 1}$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. How would the equation change if the line had a positive slope?
2. What is the equation of a line parallel to this one that passes through the point $(0, -2)$?
3. How would you find the x-intercept of this line?
4. How can you determine if a point lies on this line?
5. What is the equation of a line perpendicular to this one that passes through the origin?

Tip: Remember that the slope-intercept form $y = mx + b$ is useful for identifying both the slope and the y-intercept directly from an equation.