Question

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. $square\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$ $square18x + 20+30x=15 - 6x$ $square18x + 20+x=15 - 6x$ $square24x+30x=- 5$ $square12x + 30x=-5$

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 $\frac{3}{5}x+\frac{5}{5}x=\frac{3 + 5}{5}x=\frac{8}{5}x$). 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

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

Step3: Simplify the equation from Step 2

Combine like - terms on the left - hand side of $18x + 20+30x=15 - 6x$.
$(18x+30x)+20=15 - 6x$, $48x+20 = 15 - 6x$.
Subtract 20 from both sides: $48x=15 - 20-6x$, $48x=-5 - 6x$.
Add $6x$ to both sides: $48x + 6x=-5$, $54x=-5$.
The equations $24x + 30x=-5$ and $12x+30x=-5$ are not equivalent to the original equation.

Answer:

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