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 = "  " ; which is:  " two-thirds " .
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Explanation:
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Given:  " " ;  Solve for "x" ; 
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First, multiply the first fraction by "" ;  
   { since: " = 1" ; and since any value, multiplied by "1" results in that exact same value.  By multiplying the first fraction by:                   "" , we can get the "denominator" in the first fraction equal to the "denominator" of the second fraction.

→ " .
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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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    ""  ;  and we can rewrite the entire equation:
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→  " = \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 = "  " ;  which is "two-thirds" ; 
         
                                            →  which is:  Answer choice:  [C]:  " two-thirds" .
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