Question

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?

Answer 1

The correct answer is:  [B]:  " -4x + 2 " .
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Explanation:
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Method 1)
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Given the equation for a line:
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  "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).
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Note:  All the FOUR (4) answer choices given are written in "slope-intercept form".
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We want to convert this equation to, and rewrite this equation in,
"slope-intercept form" ;
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which is:
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" 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".
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So, given:   "y – 6 = –4(x + 1) " ;
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Let us start by expanding the "left-hand side of the equation" ;
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 that is:   " - 4(x + 1) " ;
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Note the "distributive property" of multiplication:
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a(b + c) = ab + ac ;a(b – c) = ab – ac .
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→  So;  -4(x + 1) =  -4*x  + (-4)*(1)  =  -4x + (-4)  = -4x  –  4 ;

→ Rewrite the equation:
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y – 6 = –4(x + 1) =
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   → y – 6  = -4x – 4 ;
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           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" ;
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       →  y – 6  + 6  = -4x – 4 + 6 ;
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to get:
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           →  y = -4x + 2 ; which is: Answer choice: [B] .
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Method 2).
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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:
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 "  y – 6 = –4(x + 1) " ;
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  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:
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 y – 6 + 6  = -4x – 4 + 6 ;
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 to get:
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          y = -4x + 2  ; which is: Answer choice: [B] .
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