To solve the problem of finding the equation of a line that passes through the points $(2, 6)$ and $(4, 7)$ in slope-intercept form ($y = mx + b$), let us proceed step-by-step.

Step 1: Find the slope ($m$)

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute $(x_1, y_1) = (2, 6)$ and $(x_2, y_2) = (4, 7)$: $m = \frac{7 - 6}{4 - 2} = \frac{1}{2}$

So, the slope $m = \frac{1}{2}$.

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

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using the point $(2, 6)$ and $m = \frac{1}{2}$, we substitute: $y - 6 = \frac{1}{2}(x - 2)$ Simplify: $y - 6 = \frac{1}{2}x - 1$ Add 6 to both sides: $y = \frac{1}{2}x + 5$

Final Answer:

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

Would you like further details or have any questions?

Related Questions:

1. How do you derive the slope formula for a line passing through two points?
2. What is the geometric interpretation of slope in a line?
3. How can you find the equation of a line if only its slope and one point are given?
4. How do you check if a given point lies on a particular line?
5. What changes in the equation of a line if the slope is zero?

Tip:

Always simplify fractions when finding slopes and ensure to use the proper point when substituting into the equation.