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
Verified
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.
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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