First we must 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)(-0.2) - color(blue)(0.5))/(color(red)(10) - color(blue)(3)) = -0.7/7 = -7/70 = -1/10 = -0.1#
Next, we can write an equation for the line using the point-slope formula. 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 from the problem we can write:
#(y - color(red)(0.5)) = color(blue)(-0.1)(x - color(red)(3))#
To put this into slope-intercept form we can solve for #y#. 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)(0.5) = color(blue)(-0.1)(x - color(red)(3))#
#y - color(red)(0.5) = (color(blue)(-0.1) xx x) - (color(blue)(-0.1) xx color(red)(3))#
#y - color(red)(0.5) = -0.1x + 0.3#
#y - color(red)(0.5) + 0.5 = -0.1x + 0.3 + 0.5#
#y - 0 = -0.1x + 0.8#
#y = color(red)(-0.1)x + color(blue)(0.8)#
