Answer

Verified

446.1k+ 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.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

What percentage of the solar systems mass is found class 8 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE