Standard form

Question

My algebra book doesn't really explain things clearly. It just gives me step by step answers but doesn't explain how they got this or that. In my book it's telling me how to write an equation in standard form of a line. It says "Write an equation in point slope form. Use (1,1) The slope is -3) y-1=-3(x - 1) then it says "Rewrite the equation in standard form" 3x+y=4. I understand these parts, but I have no idea how they got 4 cause it doesn't explain. 

Answer 1

The Standard Form of the equation of a line looks like:   Ax + By = C

    By convention, the "A" positive, so you may have to multiply by (-1) to make that happen.

 

The slope-intercept form of the equation of a line looks like:    y= mx + b

     where m is the slope   and b is the y-intercept

 

The point-slope form of the equation of a line looks like:

       (y - y₁) = m (x - x₁)     where m is the slope  (y₂- y₁) / (x₂ - x₁)

       Slope, m,  is sometimes called "rise" over "run" because it is the increase in the y-values divided by the increase in the x-values.  Note that a positive slope means that the line slants upward to the right and a negative slope means that the line slants downward to the right.

      Putting this together, the point-slope form is:

        (y - y₁) = ( (y₂ - y₁) / (x₂ - x₁) ) (x - x₁)

 

The problem gives the point (1,1) and a slope, m=(-3)

In point-slope form, that is:

         (y - 1) = (-3)(x - 1)

 

To get that equation into Standard Form (see above):

         (y - 1) = (-3)(x - 1)

         y - 1 = -3x + 1                [distribute]

         y = -3x + 4              [add 1 to both sides]

     3x + y = 4                    [add 3x to both sides; Standard Form]

 

Your example:

    y-3 = 3(x+1)             [point (-1,3) with slope of 3]

    y - 3 = 3x + 3              [distribute]

    y = 3x + 6              [add 3 to both sides]

   -3x + y = 6            [subtract 3x from both sides]

   3x - y = -6          [multiply by (-1) to make 3x positive]

 

Answer 2

Start with point slope form  y - 1 = -3(x - 1)  

Distribute the (-3) to get    y - 1 = -3x + 3

Add 3x to each side to get  3x + y - 1 = 3  

Add 1 to each side to get   3x + y = 4