Which equation describes the same line as y – 6 = –4(x + 1)?
a. y = –4x + 3
b. y = –4x + 2
c. y = –4x + 7
d. y = –4x – 4?

The correct answer is:  [B]:  " -4x + 2 " .

Explanation:

Method 1)

Given the equation for a line:

  "y – 6 = –4(x + 1) " ;  we are asked to choose
                    which equation  (from the answer choices provided) the describes the same line (on a graph); or, in other words,
        the same aforementioned equation,
        which is "given" (in this very question being asked).

Note:  All the FOUR (4) answer choices given are written in "slope-intercept form".

We want to convert this equation to, and rewrite this equation in,
"slope-intercept form" ;

which is:

" y = mx + b " ;
in which:"y" is a single, "stand-alone" variable on the "left-hand side of the equation"; "m" is the coefficient of "x" ;
Furthermore, "m" is the slope of the line;
"b" is the "y-intercept"; or more precisely, the value of "x"
          (that is; the "x-coordinate" of the point at which "y = 0" );
          that is, the value of "x" ; or the "x-coordinate" of the point at which
             the graph of the equation crosses the "x-axis".

So, given:   "y – 6 = –4(x + 1) " ;

Let us start by expanding the "left-hand side of the equation" ;

 that is:   " - 4(x + 1) " ;

Note the "distributive property" of multiplication:

a(b + c) = ab + ac ;a(b – c) = ab – ac .

→  So;  -4(x + 1) =  -4*x  + (-4)*(1)  =  -4x + (-4)  = -4x  –  4 ;

→ Rewrite the equation:

y – 6 = –4(x + 1) =

   → y – 6  = -4x – 4 ;

           Add "6" to EACH SIDE of the equation; to isolate "y" on the "left-hand side" of the equation ; and to write the equation in "slope-intercept form" ;

       →  y – 6  + 6  = -4x – 4 + 6 ;

to get:

           →  y = -4x + 2 ; which is: Answer choice: [B] .

Method 2).

The other formula for the equation of a line, given a particular point on that line;  
  " (x₁, y₁)" ;   is:  " y – y₁ = m(x  – x₁) " ; in which "m" is the slope of that line.

So, in the original equation in the question:

 "  y – 6 = –4(x + 1) " ;

  y₁ = 6
 m = -4

x₁ = - 1 (note: This is a NEGATIVE value, because of the right-hand side of the equation in the formula is:  "m(x – x₁)"  ;  is equal to: " –4(x + 1)" ;

So,  we and to make "x₁" negative, because "x - (-1)" = "(x+1)" ; in order to make it a positive value in the original equation.

So we now that "m = -4" ; which is the case in all the answer choices.

On the right-hand side of our equation:

  = -4(x +1) ; Use the distributive property: 
                        -4(x + 1) = -4*x  + -4(1) =  -4x – 4 ;

  y – 6 = -4x – 4    ;

Add "6" to both side of the equation; to isolate "y" on the left-hand side of the equation:

 y – 6 + 6  = -4x – 4 + 6 ;

 to get:

          y = -4x + 2  ; which is: Answer choice: [B] .
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