Question

If we have an equation as \[3\left( x-3 \right)=5\left( 2x+1 \right)\] then find \[\dfrac{{{x}^{10}}}{{{x}^{5}}}=?\]

Answer 1

Hint: In order to find the value of \[\dfrac{{{x}^{10}}}{{{x}^{5}}}\], firstly we have to simplify \[\dfrac{{{x}^{10}}}{{{x}^{5}}}\] into its simplest form. Then we have to solve the given equation i.e. \[3\left( x-3 \right)=5\left( 2x+1 \right)\] for the value of \[x\]. After finding the value of \[x\], we have to substitute its value in the simplest form of \[\dfrac{{{x}^{10}}}{{{x}^{5}}}\] and then we have to evaluate it.

Complete step-by-step solution:
Now let us know more about the linear equation in single variable. Generally, the linear equation in single variable is a equation of a straight line. The general form of a linear equation is \[ax+b=0\], where \[a\ne 0\] and \[a,b\] are real numbers. The linear equation which is a conditional equation will have only one solution. But there are cases in which the system of equations that have no solution, unique solution or no solution at all.
Now let us solve the given equation \[3\left( x-3 \right)=5\left( 2x+1 \right)\] and find the value of \[x\].
In order to solve this, we have to carry out the monomial multiplication. Upon multiplying we get,
\[\begin{align}
  & 3\left( x-3 \right)=5\left( 2x+1 \right) \\
 & 3x-9=10x+5 \\
\end{align}\]
Now let us transfer the constants to one side and the variables to the other side.
Then we get,
\[3x-10x=5+9\]
Upon further solving this we get
\[\begin{align}
  & -7x=14 \\
 & x=-2 \\
\end{align}\]
\[\therefore \] The value of \[x\] is \[-2\].
Now let us convert \[\dfrac{{{x}^{10}}}{{{x}^{5}}}\] into its simplest form.
We can convert into simplest form by applying the formula of exponents i.e. \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}},m>n\]
Upon applying this, we get
\[\dfrac{{{x}^{10}}}{{{x}^{5}}}={{x}^{10-5}}={{x}^{5}}\]
So now let us substitute the value \[x\] in \[{{x}^{5}}\]. Upon substituting we get,
\[{{x}^{5}}={{\left( -2 \right)}^{5}}=-32\]
\[\therefore \] If \[3\left( x-3 \right)=5\left( 2x+1 \right)\] then \[\dfrac{{{x}^{10}}}{{{x}^{5}}}=\] \[-32\].

Note: We must note that conditional linear equations in one variable will have only one solution. We can also solve this other method i.e. after finding the value of \[x\], instead of simplifying \[\dfrac{{{x}^{10}}}{{{x}^{5}}}\] into simplest terms, we can substitute the value and then simplify it as shown below.
We have \[x\] is \[-2\].
\[\dfrac{{{\left( -2 \right)}^{10}}}{{{\left( -2 \right)}^{5}}}=\dfrac{1024}{-32}=-32\].