Question

How do you solve $9\left( x+9 \right)=90$ using the distributive property?

Answer 1

Hint: We use the distributive property of $a\left( b+c \right)=ab+ac$ for the given numbers $a,b,c$. We solve the equation and subtract 81 both sides. Then we divide with 9 to get the solution for the variable $x$.

Complete step by step answer:
The given equation $9\left( x+9 \right)=90$ is a linear equation of $x$.
We need to solve it using the concept of distributive property.
The distributive property deals with the multiplication of a constant with the summation of two numbers.
Let’s assume the three given numbers are $a,b,c$.
Then the property tells us that $a\left( b+c \right)=ab+ac$.
We apply the property for the left-hand side of the equation $9\left( x+9 \right)=90$.
We get the variables as $a=9,b=x,c=9$.
We put the values in the equation of $a\left( b+c \right)=ab+ac$.
Putting the values, we get
$9\left( x+9 \right)=9x+81$.
Now the reformed equation becomes $9x+81=90$.
We need to solve the equation by subtracting 81
$\begin{align}
  & 9x+81-81=90-81 \\
 & \Rightarrow 9x=9 \\
\end{align}$
We now divide both sides with 9 and get $\dfrac{9x}{9}=\dfrac{9}{9}$.
Simplified form is achieved when the G.C.D of the denominator and the numerator is 1.
For our given fraction \[\dfrac{9}{9}\], the G.C.D of the denominator and the numerator is 9.
Now we divide both the denominator and the numerator with 9 and get $\dfrac{{}^{9}/{}_{9}}{{}^{9}/{}_{9}}=1$.
Therefore, the solution of $9\left( x+9 \right)=90$ is $x=1$.

Note:
We also could have formed a factorisation of the equation \[9x=9\]. We take the constant 9 common out of the reformed equation \[9x-9=0\].
Therefore, \[9x-9=9\left( x-1 \right)=0\].
The multiplication of two terms gives 0 where one of the terms is positive and non-zero. This gives that the other term has to be zero.
So, $\left( x-1 \right)=0$ which gives $x=1$ as the solution.