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

Explanation:

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

Combine $\frac{3}{5}x$ and $x$ (since $x=\frac{5}{5}x$, then $\frac{3}{5}x + x=\frac{3}{5}x+\frac{5}{5}x=\frac{8}{5}x$). The original equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$ 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 in the original equation

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

Step3: Simplify the equation from Step 2

Starting from $18x + 20+30x=15 - 6x$, combine like - terms on the left - hand side: $(18x + 30x)+20=15 - 6x$, $48x+20 = 15 - 6x$. Move all $x$ terms to one side and constants to the other side: $48x+6x=15 - 20$, $54x=-5$. The fourth option $24x + 30x=-5$ simplifies to $54x=-5$, so the fourth option is correct.

Answer:

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