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. $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$ $18x + 20+30x = 15 - 6x$ $18x + 20+x = 15 - 6x$ $24x + 30x=-5$ $12x + 30x=-5$

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 $\frac{8}{5}x+\frac{2}{3}=\frac{1}{2}-\frac{1}{5}x$.

Step2: Clear the fractions

Multiply the entire equation $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$ by the least - common multiple of the denominators (30).
\[

$$\begin{align*} 30\times(\frac{3}{5}x+\frac{2}{3}+x)&=30\times(\frac{1}{2}-\frac{1}{5}x)\\ 30\times\frac{3}{5}x+30\times\frac{2}{3}+30\times x&=30\times\frac{1}{2}-30\times\frac{1}{5}x\\ 18x + 20+30x&=15 - 6x \end{align*}$$

\]

Answer:

The equations with the same solution as $\frac{3}{5}x+\frac{2}{3}+x=\frac{1}{2}-\frac{1}{5}x$ are:

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