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Answer:
[tex]\dfrac{2}{3}x-\dfrac{13}{12}=-\dfrac{5}{12}x[/tex]
Step-by-step explanation:
There are a couple of ways to get equivalent equations. We can collect the constants on one side of the equal sign, or we can collect the variable terms on one side of the equal sign.
Constants:
[tex]\dfrac{2}{3}x-\dfrac{5}{6}=-\dfrac{5}{12}x+\dfrac{1}{4} \qquad\text{given}\\\\ \dfrac{2}{3}x-\dfrac{10}{12}-\dfrac{3}{12}=-\dfrac{5}{12}x \qquad\text{subtract $1/4=3/12$}\\\\ \dfrac{2}{3}x-\dfrac{13}{12}=-\dfrac{5}{12}x \qquad\text{matches last choice}[/tex]
Note that if we add 13/12 to this equation, the constant will be on the right, as in the second choice. However, that constant will be 13/12, not -7/12, so the second choice is not equivalent.
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Variable terms:
[tex]\dfrac{2}{3}x-\dfrac{5}{6}=-\dfrac{5}{12}x+\dfrac{1}{4} \qquad\text{given}\\\\ \dfrac{8}{12}x+\dfrac{5}{12}x-\dfrac{5}{6}=\dfrac{1}{4} \qquad\text{add $\dfrac{5}{12}x$}\\\\ \dfrac{13}{12}x-\dfrac{5}{6}=\dfrac{1}{4} \qquad\text{not equivalent to 3$^{\text{rd}}$ choice}[/tex]
Note that if we subtract 13/12x from both sides of this equation, the x-term will be on the right, as in the first choice. However, it will be -13/12x, not 1/4x, so the first choice is not equivalent, either.
The last offered choice is the equivalent equation.
