Answer :
To solve the given simultaneous equation: [tex]\(3^x \times 9^{y+1} = 181\)[/tex], we need to simplify and solve for [tex]\(x\)[/tex].
### Step-by-Step Solution:
1. Rewrite the equation:
- We know that [tex]\(9\)[/tex] can be expressed as [tex]\(3^2\)[/tex]. Hence, [tex]\(9^{y+1}\)[/tex] can be rewritten using the base 3:
[tex]\[
9^{y+1} = (3^2)^{y+1} = 3^{2(y+1)} = 3^{2y+2}
\][/tex]
2. Combine the powers of 3:
- Substitute [tex]\(9^{y+1}\)[/tex] in the original equation:
[tex]\[
3^x \times 3^{2y+2} = 181
\][/tex]
- Since the bases are the same (3), we can add the exponents:
[tex]\[
3^{x + 2y + 2} = 181
\][/tex]
3. Solve for the exponent:
- To find [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex], we take the logarithm base 3 of both sides. This gives us:
[tex]\[
x + 2y + 2 = \log_3(181)
\][/tex]
4. Express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
- Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \log_3(181) - 2y - 2
\][/tex]
Now, we've expressed [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]. This means for a given value of [tex]\(y\)[/tex], you can calculate the corresponding [tex]\(x\)[/tex] using this equation. This concludes the step-by-step solution to the problem.
### Step-by-Step Solution:
1. Rewrite the equation:
- We know that [tex]\(9\)[/tex] can be expressed as [tex]\(3^2\)[/tex]. Hence, [tex]\(9^{y+1}\)[/tex] can be rewritten using the base 3:
[tex]\[
9^{y+1} = (3^2)^{y+1} = 3^{2(y+1)} = 3^{2y+2}
\][/tex]
2. Combine the powers of 3:
- Substitute [tex]\(9^{y+1}\)[/tex] in the original equation:
[tex]\[
3^x \times 3^{2y+2} = 181
\][/tex]
- Since the bases are the same (3), we can add the exponents:
[tex]\[
3^{x + 2y + 2} = 181
\][/tex]
3. Solve for the exponent:
- To find [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex], we take the logarithm base 3 of both sides. This gives us:
[tex]\[
x + 2y + 2 = \log_3(181)
\][/tex]
4. Express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]:
- Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \log_3(181) - 2y - 2
\][/tex]
Now, we've expressed [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]. This means for a given value of [tex]\(y\)[/tex], you can calculate the corresponding [tex]\(x\)[/tex] using this equation. This concludes the step-by-step solution to the problem.
