Given the two points we can write an equation in point-slope form and then convert it to slope-intercept form. First we must 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)(7) - color(blue)(3))/(color(red)(1) - color(blue)(-1))#
#m = (color(red)(7) - color(blue)(3))/(color(red)(1) + color(blue)(1))#
#m = 4/2 = 2#
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 now use the slope we calculated and the second point to write an equation in the point-slope form:
#(y - color(red)(7)) = color(blue)(2)(x - color(red)(1))#
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. We can solve our equation for #y# to transform it to this form.
#y - color(red)(7) = (color(blue)(2) xx x) - (color(blue)(2) xx color(red)(1))#
#y - color(red)(7) = 2x - 2#
#y - color(red)(7) + 7 = 2x - 2 + 7#
#y - 0 = 2x + 5#
#y = color(red)(2)x + color(blue)(5)#
