Question

How do you solve \[a\left( n-3 \right)+8=bn\] for n?

Answer 1

Hint:In the given question, we have been asked to solve the equation for the value of ‘n’. We assume that the given equation has only one variable i.e. ‘n’ and other all are constant. To solve this question we need to get ‘n’ on one side of the “equals” sign, and all the other numbers on the other side. To solve this equation for a given variable ‘n’, we have to undo the mathematical operations such as addition, subtraction, multiplication and division that have been done to the variables. For example- in the given equation we have ‘bn’ on the right-hand side, we can easily see that a number b is multiplied to ‘n’, so we undo this step by dividing ‘b’ from the whole equation and this manner we get the solution of the question.

Complete step by step solution:
We have given that,
\[a\left( n-3 \right)+8=bn\]
Dividing both the sides of the equation by ‘b’, we get
\[\Rightarrow \dfrac{a\left( n-3 \right)}{b}+\dfrac{8}{b}=n\]
Simplifying the above equation we get
\[\Rightarrow \dfrac{an-3a}{b}+\dfrac{8}{b}=n\]
\[\Rightarrow \dfrac{an}{b}-\dfrac{3a}{b}+\dfrac{8}{b}=n\]
Subtracting \[\dfrac{an}{b}\]from both the sides of the equation, we get
\[\Rightarrow \dfrac{an}{b}-\dfrac{3a}{b}+\dfrac{8}{b}-\dfrac{an}{b}=n-\dfrac{an}{b}\]
Simplifying the above, we get
\[\Rightarrow -\dfrac{3a}{b}+\dfrac{8}{b}=n-\dfrac{an}{b}\]
Solving the right side of the equation by taking LCM, we get
\[\Rightarrow -\dfrac{3a}{b}+\dfrac{8}{b}=\dfrac{bn}{b}-\dfrac{an}{b}\]
\[\Rightarrow \dfrac{-3a+8}{b}=\dfrac{bn-an}{b}\]
Multiplying both the sides of the equation by ‘b’, we get
\[\Rightarrow -3a+8=bn-an\]
Taking ‘n’ common from the right side of the equation, we get
\[\Rightarrow -3a+8=n\times \left( b-a \right)\]
Divide both the sides of the equation by \[\left( b-a \right)\], we get
\[\Rightarrow \dfrac{-3a+8}{\left( b-a \right)}=n\]
Therefore,
\[\Rightarrow n=\dfrac{-3a+8}{\left( b-a \right)}\]
Thus, the possible value of \[n=\dfrac{-3a+8}{\left( b-a \right)}\].
It is the required solution.

Note: In the given question, no mathematical formula is being used; only the mathematical operations such as addition, subtraction, multiplication and division is used.
Use addition or subtraction properties of equality to gather variable terms on one side of the equation and constant on the other side of the equation. Use the multiplication or division properties of equality to form the coefficient of the variable term equivalent to 1. The important thing to recollect about any equation is that the ‘equals’ sign represents a balance. What the sign says is that what’s on the left-hand side is strictly an equal to what’s on the right-hand side. It is the type of question where only mathematical operations such as addition, subtraction, multiplication and division is used.