Question

Solve the equation $7\left(n-1\right)=-2\left(3+n\right)$. Also check your solution

Short answer

On solving the equation $7\left(n-1\right)=-2\left(3+n\right)$, the obtained value is $n=\frac{1}{9}$.

Step-by-step solution

Step 1. Apply the concept of Variables and Constants.

Variables: - In any expression or equation, the variable is an unknown quantity. It may be any alphabet or any special symbol.

In $7\left(n-1\right)=-2\left(3+n\right)$, n is the variable which is the unknown quantity.

Constants: - In any expression or equation, constants are the values that are fixed.

Step 2. Concept of addition.

While performing addition in an equation, the same kind of elements or like terms are added together.

In the given equation, terms with the variable n will be added all together and while performing addition constant terms which are multiplied with the variables are added.

Step 3. Introduction to the distributive property.

The distributive law of multiplication over addition or subtraction is given in below.

$\left(b\pm c\right)a=ba\pm ca$

Here $a,b$and$c$ are real numbers or variables respectively.

Step 4. Evaluation of the given equation.

$7\left(n-1\right)=-2\left(3+n\right)$

Apply distributive law in $7\left(n-1\right)$and $-2\left(3+n\right)$. Then simplify,

$\begin{array}{l}7\left(n-1\right)=-2\left(3+n\right)\\ 7n-7=-6-2n\end{array}$

Add $2n+7$in both the sides of the equation and simplify,

$\begin{array}{l}7n-7+2n+7=-6-2n+2n+7\\ 7n+2n-7+7=-6+0+7\\ 9n=1\end{array}$

Multiply both sides by $\frac{1}{9}$and simplify.

$\begin{array}{l}\frac{1}{9}\times 9n=\frac{1}{9}\times 1\\ n=\frac{1}{9}\end{array}$

Hence on solving $7\left(n-1\right)=-2\left(3+n\right)$the obtained value of n is $\frac{1}{9}$.

Step 5. Check the obtained solution.

Put $n=\frac{1}{9}$in $7\left(n-1\right)=-2\left(3+n\right)$and check whether the value on the right-hand side is equal to the value on the left-hand side.

$7\left(\left(\frac{1}{9}\right)-1\right)=-2\left(3+\left(\frac{1}{9}\right)\right)$

First, evaluate the value on left-hand side.

$\begin{array}{c}7\left(\left(\frac{1}{9}\right)-1\right)=7\left(\frac{1-9\left(1\right)}{9}\right)\\ =7\left(\frac{-8}{9}\right)\\ =-\frac{56}{9}\end{array}$

Now evaluate the values of right- hand side, that is, $-2\left(3+\left(\frac{1}{9}\right)\right)$.

$\begin{array}{c}-2\left(3+\left(\frac{1}{9}\right)\right)=-2\left(\frac{3\left(9\right)+1}{9}\right)\\ =-2\left(\frac{28}{9}\right)\\ =-\frac{56}{9}\end{array}$

Since the values of both sides are equal, the obtained result is true and verified.