To find the equation of a line passing through two points, we use the slope-intercept form of the line equation:

$y = mx + b$

Step 1: Find the slope ($m$)

The slope formula is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the given points $(6, 5)$ and $(7, 7)$:

$m = \frac{7 - 5}{7 - 6} = \frac{2}{1} = 2$

So, the slope of the line is $m = 2$.

Step 2: Use point-slope form

The point-slope form of a line is:

$y - y_1 = m(x - x_1)$

Using the point $(6, 5)$ and the slope $m = 2$:

$y - 5 = 2(x - 6)$

Simplify:

$y - 5 = 2x - 12$

$y = 2x - 12 + 5$

$y = 2x - 7$

Final Equation

The equation of the line is:

$y = 2x - 7$

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Relative Questions:

1. How would the equation change if the slope was negative?
2. What is the intercept of the line $y = 2x - 7$?
3. How can we verify that both points lie on the derived line equation?
4. What is the slope of a line perpendicular to $y = 2x - 7$?
5. What is the equation of a line parallel to $y = 2x - 7$ but passing through $(1, 3)$?

Tip:

When solving for a line equation, always double-check calculations by substituting the points back into the equation to ensure accuracy.