The correct answer is: [C]: " two-thirds " .
____________________________________________________
" x = " [tex] \frac{2}{3} [/tex] " ; which is: " two-thirds " .
____________________________________________________
Explanation:
____________________________________________________
Given: " [tex] \frac{6x + 1}{7} = \frac{18x-2}{14} [/tex] " ; Solve for "x" ;
____________________________________________________
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]" .
___________________________________________________________
Note the "distributive property of multiplication" :
_______________________________________________________
a(b + c) = ab + ac ;
a(b – c) = ab – ac .
_______________________________________________________
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:
________________________________________
"[tex] \frac{12x+2}{14} [/tex]" ; and we can rewrite the entire equation:
________________________________________
→ " [tex] \frac{12x+2}{14} [/tex] = \frac{18x-2}{14} [/tex] " ;
Let us simplify this: " 12x + 2 = 18x – 2 " ; Solve for "x" ;
_____________________________________________________
→ 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" .
__________________________________________________________
