The answer is: "No; they are not parallel."
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Explanation:
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Given:
y = 6x + 9 ; AND
27x - 3y = -81 ;
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We need to rewrite:
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" 27x - 3y = -81" ; in "slope-intercept form"; that is:
" y = mx + b" ; and see if the "slope, "m" ; is the same as: "y = 6x + 9" ; or "6";
(that is, whether "m = 6"; when an equation is written is "slope-intercept form", the slope, "m", is the coefficient of "x"; and we are given an equation that is already written in: "slope-intercept form":
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" y = 6x + 9 " ; in which, the slope, "m", = "6".
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If we can write the other equation given:
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" 27x - 3y = -81 " ; in "slope-intercept form"; and "m = 6"; (the same slope), then both equations are "parallel" to each other.
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Given:
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27x - 3y = -81 ;
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Let us write this in "slope-intercept form" ;
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Let us multiply the ENTIRE equation (both sides) by "-1"; to get rid of most of the 'negatives' :
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-1* {27x - 3y = -81} ;
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to get:
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-27x + 3y = 81 ;
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Now, add "27x" to EACH SIDE of the equation:
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-27x + 3y + 27x = 81 + 27x ;
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to get: 3y = 81 + 27x ; ↔ 3y = 27x + 81;
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Now, divide the ENTIRE EQUATION (both sides) by "3"; to isolate "y" on the left-hand side of the equation; and to re-write the equation in "slope-intercept form:
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3y / 3 = (27x + 81) / 3 ;
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y = 9x + 27 ;
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So, y = mx + b ; m = "9"; which is not "6" ;
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So the answer is: "No; they are not parallel."
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