Answer
Verified
436.5k+ views
Hint: First we will first rewrite the number 81 into powers of 3 in the equation and then use the power rule that if \[{a^x} = {a^y}\], then \[x = y\]. Then we will simplify the equations to find the value of \[x\] and \[y\].
Complete step-by-step answer:
We are given that the pair of linear equations
\[{3^{x + y}} = 81{\text{ ......eq.(1)}}\]
\[{81^{x - y}} = 3{\text{ ......eq(2)}}\]
Rewriting the number 81 into powers of 3 in the equation (1), we get
\[ \Rightarrow {3^{x + y}} = {3^4}\]
Using the power rule that if \[{a^x} = {a^y}\], then \[x = y\] in the above equation, we get
\[ \Rightarrow x + y = 4\]
Subtracting the above equation by \[x\] on both sides, we get
\[
\Rightarrow x + y - x = 4 - x \\
\Rightarrow y = 4 - x{\text{ .......eq.(3)}} \\
\]
Rewriting the number 81 into powers of 3 in the equation (2), we get
\[
\Rightarrow {3^{4\left( {x - y} \right)}} = 3 \\
\Rightarrow {3^{4x - 4y}} = {3^1} \\
\]
Using the power rule that if \[{a^x} = {a^y}\], then \[x = y\] in the above equation, we get
\[ \Rightarrow 4x - 4y = 1\]
Substituting the value of \[y\] from equation (3) in the above equation, we get
\[
\Rightarrow 4x - 4\left( {4 - x} \right) = 1 \\
\Rightarrow 4x - 16 + 4x = 1 \\
\Rightarrow 8x - 16 = 1 \\
\]
Adding the above equation by 16 on both sides, we get
\[
\Rightarrow 8x - 16 + 16 = 1 + 16 \\
\Rightarrow 8x = 17 \\
\]
Dividing the above equation by 8 on both sides, we get
\[
\Rightarrow \dfrac{{8x}}{8} = \dfrac{{17}}{8} \\
\Rightarrow x = \dfrac{{17}}{8} \\
\]
Substituting the value of \[x\] in the equation (3), we get
\[
\Rightarrow y = 4 - \dfrac{{17}}{8} \\
\Rightarrow y = \dfrac{{32 - 17}}{8} \\
\Rightarrow y = \dfrac{{15}}{8} \\
\]
Writing the value of \[x\] and \[y\] into the mixed fraction, \[{\text{Quotient}}\dfrac{{{\text{Remainder}}}}{{{\text{Divisor}}}}\] to match with the options, we get
\[ \Rightarrow x = 2\dfrac{1}{8}\]
\[ \Rightarrow y = 1\dfrac{7}{8}\]
Hence, option D is correct.
Note: We can avoid the final steps of mixed form by matching the denominators with the options and finding the correct one. We know that a linear system of two equations with two variables is any system that can be written in the form. A solution to a system of equations is a value of \[x\] and a value of \[y\] that, when substituted into the equations, satisfies both equations at the same time.
Complete step-by-step answer:
We are given that the pair of linear equations
\[{3^{x + y}} = 81{\text{ ......eq.(1)}}\]
\[{81^{x - y}} = 3{\text{ ......eq(2)}}\]
Rewriting the number 81 into powers of 3 in the equation (1), we get
\[ \Rightarrow {3^{x + y}} = {3^4}\]
Using the power rule that if \[{a^x} = {a^y}\], then \[x = y\] in the above equation, we get
\[ \Rightarrow x + y = 4\]
Subtracting the above equation by \[x\] on both sides, we get
\[
\Rightarrow x + y - x = 4 - x \\
\Rightarrow y = 4 - x{\text{ .......eq.(3)}} \\
\]
Rewriting the number 81 into powers of 3 in the equation (2), we get
\[
\Rightarrow {3^{4\left( {x - y} \right)}} = 3 \\
\Rightarrow {3^{4x - 4y}} = {3^1} \\
\]
Using the power rule that if \[{a^x} = {a^y}\], then \[x = y\] in the above equation, we get
\[ \Rightarrow 4x - 4y = 1\]
Substituting the value of \[y\] from equation (3) in the above equation, we get
\[
\Rightarrow 4x - 4\left( {4 - x} \right) = 1 \\
\Rightarrow 4x - 16 + 4x = 1 \\
\Rightarrow 8x - 16 = 1 \\
\]
Adding the above equation by 16 on both sides, we get
\[
\Rightarrow 8x - 16 + 16 = 1 + 16 \\
\Rightarrow 8x = 17 \\
\]
Dividing the above equation by 8 on both sides, we get
\[
\Rightarrow \dfrac{{8x}}{8} = \dfrac{{17}}{8} \\
\Rightarrow x = \dfrac{{17}}{8} \\
\]
Substituting the value of \[x\] in the equation (3), we get
\[
\Rightarrow y = 4 - \dfrac{{17}}{8} \\
\Rightarrow y = \dfrac{{32 - 17}}{8} \\
\Rightarrow y = \dfrac{{15}}{8} \\
\]
Writing the value of \[x\] and \[y\] into the mixed fraction, \[{\text{Quotient}}\dfrac{{{\text{Remainder}}}}{{{\text{Divisor}}}}\] to match with the options, we get
\[ \Rightarrow x = 2\dfrac{1}{8}\]
\[ \Rightarrow y = 1\dfrac{7}{8}\]
Hence, option D is correct.
Note: We can avoid the final steps of mixed form by matching the denominators with the options and finding the correct one. We know that a linear system of two equations with two variables is any system that can be written in the form. A solution to a system of equations is a value of \[x\] and a value of \[y\] that, when substituted into the equations, satisfies both equations at the same time.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which are the Top 10 Largest Countries of the World?
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE