Can you find the value of y? 36 x 2x=864 2x+2y=50

1. Break Down the Problem

We have two equations to solve for $y$:

1. $36 \times 2^x = 864$
2. $2x + 2y = 50$

2. Relevant Concepts

We will solve the first equation for $x$ and substitute it into the second equation to find $y$.

3. Analysis and Detail

Step 1: Solve the first equation for $x$

Start with the first equation: $36 \times 2^x = 864$

To isolate $2^x$: $2^x = \frac{864}{36}$

Calculating the right side: $\frac{864}{36} = 24$ So we have: $2^x = 24$

To find $x$, we take the logarithm base 2: $x = \log_2(24)$

Step 2: Use the properties of logarithms Using the change of base formula, we write: $\log_2(24) = \frac{\log_{10}(24)}{\log_{10}(2)}$

You can compute $\log_{10}(24)$ and $\log_{10}(2)$ using a scientific calculator. Assuming: $\log_{10}(24) \approx 1.3802 \quad \text{and} \quad \log_{10}(2) \approx 0.3010$ Thus, $x \approx \frac{1.3802}{0.3010} \approx 4.583$

Step 3: Solve the second equation for $y$ Now we substitute $x$ into the second equation: $2x + 2y = 50$ Substituting $x$: $2(4.583) + 2y = 50$ Calculating the first term: $9.166 + 2y = 50$ Now, isolate $2y$: $2y = 50 - 9.166$ Calculating right side: $2y = 40.834$ Finally, solve for $y$: $y = \frac{40.834}{2} = 20.417$

4. Verify and Summarize

Verify the calculations:

1. For $x \approx 4.583$, confirming $2^{4.583} \approx 24$ checks out.
2. Substituting $x$ back into the second equation gives $y \approx 20.417$.

Final Answer

The value of $y$ is approximately $20.417$.