How do you solve $9w - 4(w - 2) = 3(w + 1) - 9$?

Hint:The given equation is a linear equation in one variable $w$. To solve the above equation, we need to find the value of $w$. We have to apply the Distributive property of multiplication in the given equation to solve and simplify.

Formula used: $a(b + c) = ab + ac$

Complete step by step solution:
We have to solve $9w - 4(w - 2) = 3(w + 1) - 9$ for the value of $w$.
Distributive property of multiplication states that,
$a(b + c) = ab + ac$
In the given equation $9w - 4(w - 2) = 3(w + 1) - 9$ we apply distributive property of multiplication to the expression $4(w - 2)$ in the Left Hand Side or LHS and the expression $3(w + 1)$ in the Right Hand Side or RHS.
$ \Rightarrow 4(w - 2) = 4w - 8$and $3(w + 1) = 3w + 3$
Thus, we get:
$
9w - (4w - 4 \times 2) = (3w + 3 \times 1) - 9 \\
\Rightarrow 9w - (4w - 8) = (3w + 3) - 9 \\
$
Now we can simplify the above equation to shift all the terms with the variable $w$ in the LHS and all other terms in the RHS.
$
\Rightarrow 9w - 4w - ( - 8) = 3w + 3 - 9 \\
\Rightarrow 9w - 4w + 8 = 3w + 3 - 9 \\
$
Taking $3w$ from the RHS to the LHS and $8$ from the LHS to the RHS, we get:
$
\Rightarrow 9w - 4w - 3w = 3 - 9 - 8 \\
\Rightarrow 2w = - 14 \\
$
Thus, we see that
$
2w = - 14 \\
\Rightarrow w = \dfrac{{ - 14}}{2} = - 7 \\
$
Hence, the solution of the given equation is $w = - 7$.

Additional Information: Distributive property of multiplication means to multiply the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Note: We can check whether our solution is correct or not by putting the resulting value of $w$ in the original equation and comparing LHS with RHS. Putting $w = - 7$ in LHS we get,
$
9w - 4(w - 2) \\
= 9 \times ( - 7) - 4(( - 7) - 2) \\
= - 63 - 4( - 9) \\
= - 63 + 36 \\
= - 27 \\
$
Putting $w = - 7$ in RHS we get,
$
3(w + 1) - 9 \\
= 3( - 7 + 1) - 9 \\
= 3( - 6) - 9 \\
= - 18 - 9 \\
= - 27 \\
$
Here, LHS=RHS=$ - 27$. Since LHS = RHS, we can say that our solution of $w = - 7$ satisfies the equation and thus, the correct answer.