First, we need 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 points in the problem gives:
#m = (color(red)(-4) - color(blue)(2))/(color(red)(-7) - color(blue)(5)) = (-6)/-12 = 1/2#
Now, we can use the point-slope formula to write an equation for the line going through the two points. 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.
Substituting the slope we calculated and the first point gives:
#(y - color(red)(2)) = color(blue)(1/2)(x - color(red)(5))#
We can also substitute the slope we calculated and the second point giving:
#(y - color(red)(-4)) = color(blue)(1/2)(x - color(red)(-7))#
#(y + color(red)(4)) = color(blue)(1/2)(x + color(red)(7))#
Or, we can solve for #y# to put the equation in slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.
#y + color(red)(4) = (color(blue)(1/2) xx x) + (color(blue)(1/2) xx color(red)(7))#
#y + color(red)(4) = 1/2x + 7/2#
#y + color(red)(4) - 4 = 1/2x + 7/2 - 4#
#y + 0 = 1/2x + 7/2 - (2/2 xx 4)#
#y = 1/2x + 7/2 - 8/2#
#y = color(red)(1/2)x - color(blue)(1/2)#
