# If \[\dfrac{{10 - 3x}}{{5 + 2x}} = \dfrac{m}{n}\]; make x as the subject of the formula. Hence, find the value of x, if 3m - 4n = 2 and n = 2.5.

(a). \[x = \dfrac{{5(2n - m)}}{{2m + 3n}}\] and \[x = \dfrac{{10}}{{31}}\]

(b). \[x = \dfrac{{2(2n - m)}}{{7m + 9n}}\] and \[x = \dfrac{1}{{21}}\]

(a). \[x = \dfrac{{7(n - m)}}{{2m + n}}\] and \[x = \dfrac{{11}}{{14}}\]

(a). \[x = \dfrac{{12(n - m)}}{{2m + 3n}}\] and \[x = \dfrac{{12}}{{31}}\]

Answer

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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.

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