First, to find the slope and y-intercept we can convert this equation into the slope-intercept form by solving 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.
#x - 3y + 7 + color(red)(3y) = 0 + color(red)(3y)#
#x - 3y + color(red)(3y) + 7 = color(red)(3y)#
#x - 0 + 7 = color(red)(3y)#
#x + 7 = 3y#
or
#3y = x + 7#
#(3y)/color(red)(3) = (x + 7)/color(red)(3)#
#(color(red)(cancel(color(black)(3)))y)/cancel(color(red)(3)) = x/3 + 7/3#
#y = 1/3x + 7/3#
Therefore:
The slope is #color(red)(m = 1/3)#
The y-intercept is #color(blue)(b = 7/3)# or (0, 7/3)
To find the x-intercept we set #y = 0# and solve for #x#:
#0 = 1/3x + 7/3#
#0 - color(red)(7/3) = 1/3x + 7/3 - color(red)(7/3)#
#-7/3 = 1/3x + 0#
#-7/3 = 1/3x#
#color(red)(3) xx -7/3 = color(red)(3) xx 1/3x#
#cancel(color(red)(3)) xx -7/color(red)(cancel(color(black)(3))) = cancel(color(red)(3)) xx 1/color(red)(cancel(color(black)(3)))x#
#-7 = x# or #x = -7#
Therefore:
The x-intercept is #color(green)(x = -7)# or (-7, 0)
