How do you write the equation of a line given (5,1), (8,-2)?

First, use the two points to determine the slope of the line:

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 problem gives:

#m = (color(red)(-2) - color(blue)(1))/(color(red)(8) - color(blue)(5))#

#m = -3/3#

#m = -1#

We can now use the point-slope formula using either point from the problem and the slope we calculated to determine the equation of the line.

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.

Substitution gives:

#(y - color(red)(-2)) = color(blue)(-1)(x - color(red)(8))#

#(y + color(red)(2)) = color(blue)(-1)(x - color(red)(8))#

We can solve for #y# to put this equation into the more familiar slope-intercept form:

#y + color(red)(2) = color(blue)(-1)x - (color(blue)(-1) xx color(red)(8))#

#y + color(red)(2) = -x - (-8)#

#y + color(red)(2) = -x + 8#

#y + color(red)(2) - 2 = -x + 8 - 2#

#y + 0 = -x + 6#

#y = -x + 6#