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)(3) - color(blue)(0))/(color(red)(10) - color(blue)(6)) = 3/4#
Point Slope
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 values from the first point gives:
#(y - color(red)(0)) = color(blue)(3/4)(x - color(red)(6))#
We can also substitute the slope we calculated and the values from the second point giving:
#(y - color(red)(3)) = color(blue)(3/4)(x - color(red)(10))#
Slope Intercept
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 take the first point-slope equation and solve for #y#:
#y - color(red)(0) = color(blue)(3/4)(x - color(red)(6))#
#y = (color(blue)(3/4) xx x) - (color(blue)(3/4) xx color(red)(6))#
#y = 3/4x - 18/4#
#y = color(red)(3/4)x - color(blue)(9/2)#
Standard Form
The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)#
Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
We can convert the slope intercept to Standard Form as follows:
#y = color(red)(3/4)x - color(blue)(9/2)#
#-3/4x + y = -3/4x + color(red)(3/4)x - color(blue)(9/2)#
#-3/4x + y = 0 - color(blue)(9/2)#
#-3/4x + y = -color(blue)(9/2)#
#color(red)(-4)(-3/4x + y) = color(red)(-4) xx -color(blue)(9/2)#
#3x - 4y = 36/2#
#color(red)(3)x + color(blue)(-4)y = color(green)(18)#
Domain and Range
There are no restrictions on the value of #x#, therefore the Domain is: The set of of Real Numbers or #{RR}#
From the slope-intercept form, as #x# increases, #y# increases, as #x# decreases, #y# decreases. Therefore the Range is: The set of of Real Numbers or #{RR}#
