Answer

Verified

485.7k+ views

Hint: Solve the given equation to find x in terms of m and n. Then, solve for m and n from the given equations and find them. Then, substitute these values of m and n in the given equation to solve and find x.

Let us represent x in terms of m and n, starting from the given equation.

\[\dfrac{{10 - 3x}}{{5 + 2x}} = \dfrac{m}{n}\]

Cross-multiplying, we get:

\[n(10 - 3x) = m(5 + 2x)\]

Multiplying m and n inside the bracket, we get:

\[10n - 3nx = 5m + 2mx\]

Gather all terms containing x on the left-hand side of the equation to get as follows:

\[ - 2mx - 3nx = 5m - 10n\]

Now, take x as a common term from the left-hand side of the equations:

\[x( - 2m - 3n) = 5m - 10n\]

Solve for x to get as follows:

\[x = \dfrac{{5m - 10n}}{{ - 2m - 3n}}\]

Now take 5 as common term from the numerator to get:

\[x = \dfrac{{5(m - 2n)}}{{ - 2m - 3n}}\]

Now multiply numerator and denominator by -1 to get the final expression.

\[x = \dfrac{{5(2n - m)}}{{2m + 3n}}..........(1)\]

Given that, n = 2.5, substitute it in the equation 3m – 4n =2 to find the value of m.

\[n = 2.5..........(2)\]

\[3m - 4(2.5) = 2\]

\[3m - 10 = 2\]

Take 10 to the other side and add it with 2 to get 12.

\[3m = 2 + 10\]

\[3m = 12\]

Solve for m as follows:

\[m = \dfrac{{12}}{3}\]

Simplifying to obtain the value of m.

\[m = 4...........(3)\]

Substitute equation (3) and equation (2) in equation (1) to get as follows:

\[x = \dfrac{{5(2(2.5) - 4)}}{{2(4) + 3(2.5)}}\]

\[x = \dfrac{{5(5 - 4)}}{{8 + 7.5}}\]

\[x = \dfrac{5}{{15.5}}\]

Multiply numerator and denominator by 2 to obtain the final expression.

\[x = \dfrac{5}{{15.5}} \times \dfrac{2}{2}\]

\[x = \dfrac{{10}}{{31}}\]

Hence, the correct answer is option (a).

Note: Even though the ratio of m and n is represented as a function of x, we can solve them to find the value of x in terms of m and n. Don’t confuse yourself with the phrase “make the subject of the equation”, it just means express x explicitly.

__Complete step-by-step answer:__Let us represent x in terms of m and n, starting from the given equation.

\[\dfrac{{10 - 3x}}{{5 + 2x}} = \dfrac{m}{n}\]

Cross-multiplying, we get:

\[n(10 - 3x) = m(5 + 2x)\]

Multiplying m and n inside the bracket, we get:

\[10n - 3nx = 5m + 2mx\]

Gather all terms containing x on the left-hand side of the equation to get as follows:

\[ - 2mx - 3nx = 5m - 10n\]

Now, take x as a common term from the left-hand side of the equations:

\[x( - 2m - 3n) = 5m - 10n\]

Solve for x to get as follows:

\[x = \dfrac{{5m - 10n}}{{ - 2m - 3n}}\]

Now take 5 as common term from the numerator to get:

\[x = \dfrac{{5(m - 2n)}}{{ - 2m - 3n}}\]

Now multiply numerator and denominator by -1 to get the final expression.

\[x = \dfrac{{5(2n - m)}}{{2m + 3n}}..........(1)\]

Given that, n = 2.5, substitute it in the equation 3m – 4n =2 to find the value of m.

\[n = 2.5..........(2)\]

\[3m - 4(2.5) = 2\]

\[3m - 10 = 2\]

Take 10 to the other side and add it with 2 to get 12.

\[3m = 2 + 10\]

\[3m = 12\]

Solve for m as follows:

\[m = \dfrac{{12}}{3}\]

Simplifying to obtain the value of m.

\[m = 4...........(3)\]

Substitute equation (3) and equation (2) in equation (1) to get as follows:

\[x = \dfrac{{5(2(2.5) - 4)}}{{2(4) + 3(2.5)}}\]

\[x = \dfrac{{5(5 - 4)}}{{8 + 7.5}}\]

\[x = \dfrac{5}{{15.5}}\]

Multiply numerator and denominator by 2 to obtain the final expression.

\[x = \dfrac{5}{{15.5}} \times \dfrac{2}{2}\]

\[x = \dfrac{{10}}{{31}}\]

Hence, the correct answer is option (a).

Note: Even though the ratio of m and n is represented as a function of x, we can solve them to find the value of x in terms of m and n. Don’t confuse yourself with the phrase “make the subject of the equation”, it just means express x explicitly.

Recently Updated Pages

Who among the following was the religious guru of class 7 social science CBSE

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

Trending doubts

Write the difference between order and molecularity class 11 maths CBSE

A rainbow has circular shape because A The earth is class 11 physics CBSE

Which are the Top 10 Largest Countries of the World?

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

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

Give 10 examples for herbs , shrubs , climbers , creepers

What are noble gases Why are they also called inert class 11 chemistry CBSE

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

Differentiate between calcination and roasting class 11 chemistry CBSE