Question

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 $x = \dfrac{{12}}{{31}}$

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.