use properties to rewrite the given equation. which equations have the same solution as the equation…

use properties to rewrite the given equation. which equations have the same solution as the equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$. select three options. $square\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$ $square18x + 20+30x = 15 - 6x$ $square18x + 20+x = 15 - 6x$ $square24x + 30x=-5$ $square12x + 30x=-5$

use properties to rewrite the given equation. which equations have the same solution as the equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$. select three options. $square\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$ $square18x + 20+30x = 15 - 6x$ $square18x + 20+x = 15 - 6x$ $square24x + 30x=-5$ $square12x + 30x=-5$

Answer

Answer:

A. $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$, B. $18x + 20+30x=15 - 6x$, D. $24x+30x=- 5$

Explanation:

Step1: Combine like - terms on the left - hand side of the original equation

Combine $\frac{3}{5}x$ and $x$ in $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$. Since $x=\frac{5}{5}x$, then $\frac{3}{5}x+\frac{5}{5}x=\frac{8}{5}x$, and the equation becomes $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$, so the first option is correct.

Step2: Clear the fractions

Multiply the original equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$ by 30 (the least common multiple of 5, 3, and 2). $30\times(\frac{3}{5}x)+30\times\frac{2}{3}+30\times x = 30\times\frac{1}{2}-30\times\frac{1}{5}x$. $18x + 20+30x=15 - 6x$, so the second option is correct.

Step3: Simplify the equation from Step2

Starting from $18x + 20+30x=15 - 6x$, first combine like - terms on the left - hand side: $48x+20 = 15 - 6x$. Then move all $x$ terms to one side and constants to the other side: $48x+6x=15 - 20$. $54x=-5$. Also, from $18x + 20+30x=15 - 6x$, we can rewrite it as $24x+30x=-5$ (by subtracting $6x$ from both sides and subtracting 20 from both sides), so the fourth option is correct.