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Question: How do you solve \(9w - 4(w - 2) = 3(w + 1) - 9\)?...

How do you solve 9w−4(w−2)=3(w+1)−99w - 4(w - 2) = 3(w + 1) - 9?

Explanation

Solution

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

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

Complete step by step solution:
We have to solve 9w−4(w−2)=3(w+1)−99w - 4(w - 2) = 3(w + 1) - 9 for the value of ww.
Distributive property of multiplication states that,
a(b+c)=ab+aca(b + c) = ab + ac
In the given equation 9w−4(w−2)=3(w+1)−99w - 4(w - 2) = 3(w + 1) - 9 we apply distributive property of multiplication to the expression 4(w−2)4(w - 2) in the Left Hand Side or LHS and the expression 3(w+1)3(w + 1) in the Right Hand Side or RHS.
⇒4(w−2)=4w−8\Rightarrow 4(w - 2) = 4w - 8and 3(w+1)=3w+33(w + 1) = 3w + 3
Thus, we get:
9w−(4w−4×2)=(3w+3×1)−9 ⇒9w−(4w−8)=(3w+3)−9  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 ww in the LHS and all other terms in the RHS.
⇒9w−4w−(−8)=3w+3−9 ⇒9w−4w+8=3w+3−9  \Rightarrow 9w - 4w - ( - 8) = 3w + 3 - 9 \\\ \Rightarrow 9w - 4w + 8 = 3w + 3 - 9 \\\
Taking 3w3w from the RHS to the LHS and 88 from the LHS to the RHS, we get:
⇒9w−4w−3w=3−9−8 ⇒2w=−14  \Rightarrow 9w - 4w - 3w = 3 - 9 - 8 \\\ \Rightarrow 2w = - 14 \\\
Thus, we see that
2w=−14 ⇒w=−142=−7  2w = - 14 \\\ \Rightarrow w = \dfrac{{ - 14}}{2} = - 7 \\\
Hence, the solution of the given equation is w=−7w = - 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 ww in the original equation and comparing LHS with RHS. Putting w=−7w = - 7 in LHS we get,
9w−4(w−2) =9×(−7)−4((−7)−2) =−63−4(−9) =−63+36 =−27  9w - 4(w - 2) \\\ = 9 \times ( - 7) - 4(( - 7) - 2) \\\ = - 63 - 4( - 9) \\\ = - 63 + 36 \\\ = - 27 \\\
Putting w=−7w = - 7 in RHS we get,
3(w+1)−9 =3(−7+1)−9 =3(−6)−9 =−18−9 =−27  3(w + 1) - 9 \\\ = 3( - 7 + 1) - 9 \\\ = 3( - 6) - 9 \\\ = - 18 - 9 \\\ = - 27 \\\
Here, LHS=RHS=−27 - 27. Since LHS = RHS, we can say that our solution of w=−7w = - 7 satisfies the equation and thus, the correct answer.