The correct answer is:
[C]: "
two-thirds <span>" .
</span>
____________________________________________________"
x = "
![\frac{2}{3}](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B3%7D%20)
" ; which is: "
two-thirds " .
____________________________________________________Explanation:____________________________________________________Given: "
![\frac{6x + 1}{7} = \frac{18x-2}{14}](https://tex.z-dn.net/?f=%20%5Cfrac%7B6x%20%2B%201%7D%7B7%7D%20%3D%20%5Cfrac%7B18x-2%7D%7B14%7D%20)
" ; Solve for "x" ;
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First, multiply the first fraction by "
![\frac{2}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B2%7D%20)
" ;
{ since: "
![\frac{2}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B2%7D%20)
= 1" ; and since any value, multiplied by "1" results in that exact same value. By multiplying the first fraction by: "
![\frac{2}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B2%7D%20)
" , we can get the "denominator" in the first fraction equal to the "denominator" of the second fraction.
→
![\frac{2}{2}*( \frac{6x + 1}{7})=\frac{2(6x+1)}{2(7)} = \\ "\frac{2(6x+1)}{14}](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B2%7D%2A%28%20%5Cfrac%7B6x%20%2B%201%7D%7B7%7D%29%3D%5Cfrac%7B2%286x%2B1%29%7D%7B2%287%29%7D%20%3D%20%5C%5C%20%20%20%20%22%5Cfrac%7B2%286x%2B1%29%7D%7B14%7D%20)
" .
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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:
________________________________________
"
![\frac{12x+2}{14}](https://tex.z-dn.net/?f=%20%5Cfrac%7B12x%2B2%7D%7B14%7D%20)
" ; and we can rewrite the entire equation:
________________________________________
→ "
![\frac{12x+2}{14}](https://tex.z-dn.net/?f=%20%5Cfrac%7B12x%2B2%7D%7B14%7D%20)
= \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 = "
![\frac{2}{3}](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B3%7D%20)
" ; which is "
two-thirds" ;
→ which is:
Answer choice: [C]: "<span>
two-thirds</span><span>" .
</span>
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