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)(11) - color(blue)(2))/(color(red)(-23) - color(blue)(30)) = 9/-53 = -9/53#
We can now use the point-slope formula to find an equation for the line between the two points. The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#
Where #(color(blue)(x_1), color(blue)(y_1))# is a point on the line and #color(red)(m)# is the slope.
Substituting the slope we calculated and the values from the first point in the problem gives:
#(y - color(blue)(2)) = color(red)(-9/53)(x - color(blue)(30))#
We can also substitute the slope we calculated and the values from the second point in the problem gives:
#(y - color(blue)(11)) = color(red)(-9/53)(x - color(blue)(-23))#
#(y - color(blue)(11)) = color(red)(-9/53)(x + color(blue)(23))#
We can also solve the first equation for #y# to transform the equation to 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(blue)(2) = (color(red)(-9/53) xx x) - (color(red)(-9/53) xx color(blue)(30))#
#y - color(blue)(2) = -9/53x - (-270/53)#
#y - color(blue)(2) = -9/53x + 270/53#
#y - color(blue)(2) + 2 = -9/53x + 270/53 + 2#
#y - 0 = -9/53x + 270/53 + (53/53 xx 2)#
#y - 0 = -9/53x + 270/53 + 106/53#
#y = color(red)(-9/53)x + color(blue)(376/53)#
