What is the value of x in the proportion StartFraction 6 x plus 1 over 7 EndFraction equals StartFraction 18 x minus 2 over 14 EndFraction? A. 0 B. 3 C. two-thirds D. one-fourteenth

The correct answer is:  [C]:  " two-thirds " .
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" x = " [tex] \frac{2}{3} [/tex] " ; which is:  " two-thirds " .
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Explanation:
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Given:  " [tex] \frac{6x + 1}{7} = \frac{18x-2}{14} [/tex] " ;  Solve for "x" ; 
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First, multiply the first fraction by "[tex] \frac{2}{2} [/tex]" ;  
   { since: "[tex] \frac{2}{2} [/tex] = 1" ; and since any value, multiplied by "1" results in that exact same value.  By multiplying the first fraction by:                   "[tex] \frac{2}{2} [/tex]" , we can get the "denominator" in the first fraction equal to the "denominator" of the second fraction.

→ [tex] \frac{2}{2}*( \frac{6x + 1}{7})=\frac{2(6x+1)}{2(7)} =

    "\frac{2(6x+1)}{14} [/tex]" .
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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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As such, take the "numerator" ; which is:

" 2(6x + 1) " ;  and expand:

→  =  (2*6x) + (2*1) =  " 12x + 2 " ;

Now, we can rewrite the entire expression:
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    "[tex] \frac{12x+2}{14} [/tex]"  ;  and we can rewrite the entire equation:
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→  " [tex] \frac{12x+2}{14} [/tex] = \frac{18x-2}{14} [/tex] " ;

Let us simplify this:  " 12x + 2 = 18x – 2 " ;  Solve for "x" ;
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  →  18x – 2 = 12x + 2 ; 

  →  Subtract "12x" from each side of the equation; and subtract "2" from each side of the equation:


  →  18x – 2 – 12x – 2 = 12x + 2 – 12x – 2 ; 

to get: 

       →  6x – 4 = 0  ; 

Add "4" to each side of the equation:

       →  6x – 4 + 4 = 0 + 4 ; 

to get:

       →  6x = 4  ;

Divide each side of the equation by "6" ;
  to isolate "x" on one side of the equation; & to solve for "x" ; 

→ 6x / 6  = 4 / 6  ;

→  x = 4/6 ;   → "4/6" = "(4÷2)/(6÷2)" = "2/3" .

→  x = 2/3 ;   →  x = " [tex] \frac{2}{3} [/tex] " ;  which is "two-thirds" ; 
         
                                            →  which is:  Answer choice:  [C]:  " two-thirds" .
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