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