Given two points we can use the point-slope formula to find an equation for a line.
First, determine the slope. The slope can be found by using the formula: m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))
Where m is the slope and (color(blue)(x_1, y_1)) and (color(red)(x_2, y_2)) are the two points on the line.
Substituting the values from the points in the problem gives:
m = (color(red)(3) - color(blue)(2))/(color(red)(-7) - color(blue)(5))
m = 1/-12 = -1/12
The point-slope formula states: (y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))
Where color(blue)(m) is the slope and color(red)(((x_1, y_1))) is a point the line passes through.
We can use the first point and the slope we calculate to give:
(y - color(red)(2)) = color(blue)(-1/12)(x - color(red)(5))
We can also use the second point and the slope we calculate to give:
(y - color(red)(3)) = color(blue)(-1/12)(x - color(red)(-7))
(y - color(red)(3)) = color(blue)(-1/12)(x + color(red)(7))
We can also solve this for y to give an equation in slope-intercept form:
y - color(red)(3) = (color(blue)(-1/12) xx x) + (color(blue)(-1/12) xx color(red)(7))
y - color(red)(3) = color(blue)(-1/12)x - 7/12
y - color(red)(3) + 3 = color(blue)(-1/12)x - 7/12 + 3
y - 0 = color(blue)(-1/12)x - 7/12 + 36/12
y = -1/12x + 29/12
