Question

Solve the following equations by transposition method:
(a) \[x-\dfrac{x}{4}-\dfrac{1}{2}=3+\dfrac{x}{4}\]
(b) \[14=\dfrac{7x}{10}-8\]

Answer 1

Hint: We solve these equations using the transposition method that is nothing but adding or subtracting or multiplying of dividing with certain numbers on both sides in order to reduce either of side to unknown value \['x'\] to get the answer. For example if the equation is of form
\[\Rightarrow x+a=b\]
Then we use transposition method and subtract \['a'\] on both sides we get
\[\begin{align}
  & \Rightarrow x+a-a=b-a \\
 & \Rightarrow x=b-a \\
\end{align}\]
So, we find the unknown value by using this method.

Complete step-by-step answer:
Let us solve the first equation
(a) \[x-\dfrac{x}{4}-\dfrac{1}{2}=3+\dfrac{x}{4}\]
First, let us convert the equation in such a way that unknown variable lies only one side
Now, by subtracting \[\dfrac{x}{4}\] from both sides we get
\[\begin{align}
  & \Rightarrow x-\dfrac{x}{4}-\dfrac{1}{2}-\dfrac{x}{4}=3+\dfrac{x}{4}-\dfrac{x}{4} \\
 & \Rightarrow x-\dfrac{x}{2}-\dfrac{1}{2}=3 \\
\end{align}\]
Now, by simplifying the similar terms from both sides we get
\[\Rightarrow \dfrac{x}{2}-\dfrac{1}{2}=3\]
Now, by adding \[\dfrac{1}{2}\] on both sides we get
\[\begin{align}
  & \Rightarrow \dfrac{x}{2}-\dfrac{1}{2}+\dfrac{1}{2}=3+\dfrac{1}{2} \\
 & \Rightarrow \dfrac{x}{2}=\dfrac{7}{2} \\
\end{align}\]
Now, let us multiply with 2 on both sides we get
\[\begin{align}
  & \Rightarrow \dfrac{x}{2}\times 2=\dfrac{7}{2}\times 2 \\
 & \Rightarrow x=7 \\
\end{align}\]
Therefore, the value of unknown variable in the given equation is
\[\therefore x=7\]
(b) \[14=\dfrac{7x}{10}-8\]
Here, let us convert the equation in such a way that variables lie on one side and constants lie on the other side.
So, by adding 8 on both sides we get
\[\begin{align}
  & \Rightarrow 14+8=\dfrac{7x}{10}-8+8 \\
 & \Rightarrow 22=\dfrac{7x}{10} \\
\end{align}\]
Now, let us convert the fractions into integers.
So, by multiplying with 10 on both sides we get
\[\begin{align}
  & \Rightarrow 22\times 10=\dfrac{7x}{10}\times 10 \\
 & \Rightarrow 7x=220 \\
\end{align}\]
Now, by dividing with 7 on both sides we get
\[\begin{align}
  & \Rightarrow 7x\div 7=220\div 7 \\
 & \Rightarrow x=\dfrac{220}{7} \\
\end{align}\]
Therefore, the value of unknown variable in the given equation is
\[\therefore x=\dfrac{220}{7}\]

Note: Students should keep in mind that we need to use the transposition method. The transposition method involves adding or subtracting or multiplying or dividing with certain numbers on both sides of the equation in order to reduce either side to unknown value \[‘x’\] to get the answer. This is the correct method to follow. We should not take the constant numbers from one side to other side that is if we have
\[\Rightarrow x+a=b\]
Then we directly take the number \[‘a’\] from LHS to RHS which is not a transposition method. This point has to be taken care of. We can check if our answer is correct by substituting the obtained value of x in the given equation and checking if it satisfies the equation or not.