Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Connect with professionals on our platform to receive accurate answers to your questions quickly and efficiently. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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]
[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]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
