Determine, in standard form, the equation of the straight line parallel to #3x - 6y = 6# that passes through #(4,-3)#?

#y = 1/2x - 5#

graph{x/2 - 5 [-2.75, 17.25, -6.84, 3.16]}

You can check that it passes through #(4,-3)#.

In standard form, your equation should look like #y = mx + b#, where #b# is the y-intercept and #m# is the slope.

Since the line passes through #(x,y) = (4,-3)#, and you are given

#3x - 6y = 6#,

Try solving for the standard form.

#3x = 6y + 6#

#6y = 3x - 6#

#y = 3/6x - 6/6#

#= 1/2x - 1#

Let's see if it passes through #(4,-3)#. Plug in #x = 4#:

#y = 1/2(4) - 1 = 2 - 1 = color(red)(1 ne -3)#

So it's not this equation as it is, but it's the equation PARALLEL to #y = 1/2x - 1#. Parallel means the same slope, but a different y-intercept. So, what we do is constrain the solution so that #y = -3# must give #x = 4#.

That is, we solve for the y-intercept that satisfies this constraint.

#-3 = 1/2(4) + b#

#-3 = 2 + b#

#=> b = -5#

Therefore, the equation for the line parallel to #3x - 6y = 6#, passing through #(4,-3)#, is:

#color(blue)(y = 1/2x - 5)#

As usual, it is always good to verify your answer. We check whether #(4,-3)# is passed through.

#-3 stackrel(?" ")(=) 1/2(4) - 5#

#-3 stackrel(?" ")(=) 2 - 5 = -3 color(blue)(sqrt"")#

So, yes, this is the solution.