To find the equation of the line in the form $y = mx + b$ that passes through the points $(1, 2)$ and $(7, -1)$, we can follow these steps:

Step 1: Find the slope $m$

The slope $m$ of the line is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the given points $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (7, -1)$: $m = \frac{-1 - 2}{7 - 1} = \frac{-3}{6} = -\frac{1}{2}$

Step 2: Use the point-slope form to find the y-intercept $b$

The equation of the line can now be written as: $y = -\frac{1}{2}x + b$ To find $b$, substitute one of the points into the equation. Let's use the point $(1, 2)$: $2 = -\frac{1}{2}(1) + b$ $2 = -\frac{1}{2} + b$ Add $\frac{1}{2}$ to both sides to solve for $b$: $b = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$

Final Equation

The equation of the line is: $y = -\frac{1}{2}x + \frac{5}{2}$

So, the value of $b$ is $\frac{5}{2}$.

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you calculate the slope if given different points?
2. How can you convert the equation to standard form?
3. What is the significance of the y-intercept in a linear equation?
4. How would the equation change if the points were reversed?
5. How do you interpret the slope in a real-world context?

Tip: Remember, the slope $m$ represents the rate of change of $y$ with respect to $x$, and the y-intercept $b$ is the value of $y$ when $x = 0$.