To determine which of the provided equations have the same solution as the original equation
[tex]\[
\frac{3}{5} x + \frac{2}{3} + x = \frac{1}{2} - \frac{1}{5} x,
\][/tex]
we need to carefully examine and simplify the given equation and each of the options to see which ones are equivalent. Here's a step-by-step approach:
1. Initial Equation:
[tex]\[
\frac{3}{5} x + \frac{2}{3} + x = \frac{1}{2} - \frac{1}{5} x
\][/tex]
2. Simplifying the left-hand side (LHS) of the original equation:
[tex]\[
\frac{3}{5} x + x + \frac{2}{3} = \frac{3}{5} x + \frac{5}{5} x + \frac{2}{3} = \left(\frac{3}{5} + \frac{5}{5}\right) x + \frac{2}{3} = \frac{8}{5} x + \frac{2}{3}
\][/tex]
3. Simplifying the right-hand side (RHS) of the original equation:
[tex]\[
\frac{1}{2} - \frac{1}{5} x
\][/tex]
Thus, the simplified original equation becomes:
[tex]\[
\frac{8}{5} x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5} x
\][/tex]
4. Comparison with the given options:
- Option 1:
[tex]\[
\frac{8}{5} x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5} x
\][/tex]
This matches our simplified original equation directly.
- Option 2:
[tex]\[
18 x + 20 + 30 x = 15 - 6 x
\][/tex]
Simplifying both sides:
[tex]\[
48 x + 20 = 15 - 6 x
\][/tex]
Adding [tex]\(6x\)[/tex] to both sides:
[tex]\[
54 x + 20 = 15
\][/tex]
Subtracting 20 from both sides:
[tex]\[
54 x = -5
\][/tex]
This equation doesn’t match our simplified form.
- Option 3:
[tex]\[
18 x + 20 + x = 15 - 6 x
\][/tex]
Simplifying both sides:
[tex]\[
19 x + 20 = 15 - 6 x
\][/tex]
Adding [tex]\(6x\)[/tex] to both sides:
[tex]\[
25 x + 20 = 15
\][/tex]
Subtracting 20 from both sides:
[tex]\[
25 x = -5
\][/tex]
This equation also doesn’t match our simplified form.
- Option 4:
[tex]\[
24 x + 30 x = -5
\][/tex]
Simplify the LHS:
[tex]\[
54 x = -5
\][/tex]
This equation doesn’t match our simplified form.
- Option 5:
[tex]\[
12 x + 30 x = -5
\][/tex]
Simplify the LHS:
[tex]\[
42 x = -5
\][/tex]
This equation also doesn’t match our simplified form.
From the analysis, the only equation that matches or simplifies to the original equation is Option 1:
[tex]\[
\frac{8}{5} x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5} x
\][/tex]
Therefore, the equations that have the same solution as the original equation are:
- [tex]$\frac{8}{5} x + \frac{2}{3} = \frac{1}{2} - \frac{1}{5} x$[/tex]
