Question

Solve the linear equation
\[\dfrac{{x - 5}}{3} = \dfrac{{x - 3}}{5}\]
a). 0
b). 8
c). 6
d). 9

Answer 1

Hint: We can solve this in two methods. One is by substituting the given options in the given problem and if we get LHS is equal to RHS then the substituted option is the correct answer. Another method is we solve it by cross multiplying it and by using the transposition method. We simplify this to find the value of ‘x’.

Complete step-by-step solution:
Given, \[\dfrac{{x - 5}}{3} = \dfrac{{x - 3}}{5}\].
Cross multiplying we have
\[5\left( {x - 5} \right) = 3\left( {x - 3} \right)\]
Now expanding the brackets we have,
\[5x - 25 = 3x - 9\]
We transpose \[ - 25\] which is present in the left-hand side of the equation to the right-hand side of the equation by adding \[25\]on the right-hand side of the equation.
\[5x = 3x - 9 + 25\]
Similarly we transpose 3x to the left hand side of the equation by subtracting 3x on the left hand side of the equation,
\[5x - 3x = - 9 + 25\]
We can see that the variable ‘x’ and the constants are separated, then
\[2x = 16\]
Divide the whole equation by 2
\[x = \dfrac{{16}}{2}\]
\[ \Rightarrow x = 8\].
This is the required result. Hence the correct option is (b).

Note: By simplifying we have obtained the answer for ‘p’. We can check whether the obtained value of ‘x’ is correct or not. To check we simply substitute the obtained value of ‘p’ in the given problem. If L.H.S is equal to R.H.S. then our answer is correct.
\[\dfrac{{x - 5}}{3} = \dfrac{{x - 3}}{5}\]
\[\Rightarrow \dfrac{{8 - 5}}{3} = \dfrac{{8 - 3}}{5}\]
\[\Rightarrow \dfrac{3}{3} = \dfrac{5}{5}\]
\[ \Rightarrow 1 = 1\]
Hence the obtained answer is correct.
In the above, we did the transpose of addition and subtraction. Similarly, if we have multiplication, we use division to transpose. If we have division, we use multiplication to transpose. Follow the same procedure for these kinds of problems.