Find the product xy if {:[2^(x)+3^(y)=5],[2^(x+2)+3^(y+1)=18]:}

Question

Find the product  xy if  {:[2^(x)+3^(y)=5],[2^(x+2)+3^(y+1)=18]:}

Bytelearn · Accepted Answer

Analyze Equation: Analyze the first equation 2^(x) + 3^(y) = 5. We need to find values of x and y that satisfy this equation. Since 5 is a small number, we can guess and check small integer values for x and y.Test Small Values: Test small integer values for x and y in the first equation. If x = 1, then 2^(1) = 2, and we need 3^(y) to be 3 to make the sum 5. This means y = 1. So, one possible solution is x = 1 and y = 1.Verify First Solution: Verify the solution x = 1 and y = 1 in the second equation 2^(x+2) + 3^(y+1) = 18. Substitute x = 1 into 2^(x+2) to get 2^(1+2) = 2^3 = 8. Substitute y = 1 into 3^(y+1) to get 3^(1+1) = 3^2 = 9. Now, add these two results to see if they equal 18: 8 + 9 = 17. This does not satisfy the second equation, so x = 1 and y = 1 is not the correct solution.Try Other Values: Since x = 1 and y = 1 is not a solution, we need to try other small integer values. If x = 0, then 2^(0) = 1, and we need 3^(y) to be 4 to make the sum 5. However, 3^(y) cannot be 4 for any integer y, so x cannot be 0. If x = 2, then 2^(2) = 4, and we need 3^(y) to be 1 to make the sum 5. This means y = 0. So, another possible solution is x = 2 and y = 0.Verify Second Solution: Verify the solution x = 2 and y = 0 in the second equation 2^(x+2) + 3^(y+1) = 18. Substitute x = 2 into 2^(x+2) to get 2^(2+2) = 2^4 = 16. Substitute y = 0 into 3^(y+1) to get 3^(0+1) = 3^1 = 3. Now, add these two results to see if they equal 18: 16 + 3 = 19. This does not satisfy the second equation, so x = 2 and y = 0 is not the correct solution.Final Attempt: Since x = 2 and y = 0 is not a solution, we need to try other small integer values. If x = -1, then 2^(-1) = 1/2, which is not an integer and will not help us find an integer solution for y. Therefore, x cannot be -1. If x = 2, then 2^(2) = 4, and we need 3^(y) to be 1 to make the sum 5. This means y = 0. We have already tried this combination and it did not work, so we made a mistake by considering it again.

Find the product $xy$ if $\newline$ $\begin{cases} 2^{x}+3^{y}=5 \ 2^{x+2}+3^{y+1}=18 \end{cases}$

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Find the product xyxyxy if\newline{2x+3y=5 2x+2+3y+1=18\begin{cases} 2^{x}+3^{y}=5 \ 2^{x+2}+3^{y+1}=18 \end{cases}{2x+3y=5 2x+2+3y+1=18​

Find the product $xy$ if $\newline$ $\begin{cases} 2^{x}+3^{y}=5 \ 2^{x+2}+3^{y+1}=18 \end{cases}$