Question

How do you solve \[0 = 4\left( {n - 4} \right) - 6\left( {n - 4} \right)\] using the distributive property?

Answer 1

Hint: To solve the given equation we need to apply distributive property as the given equation is linear equation and the equation consists of constant variable n, hence to solve for the given equation combine all the like terms and using the distributive property and then simplify the terms to get the value of n.

Complete step by step solution:
Let us write the given equation:
 \[0 = 4\left( {n - 4} \right) - 6\left( {n - 4} \right)\]
First, multiply each term with each parenthesis by the term outside the parentheses:
 \[0 = 4\left( {n - 4} \right) - 6\left( {n - 4} \right)\]
 \[ \Rightarrow 0 = \left( {4 \times n} \right) - \left( {4 \times 4} \right) - \left( {6 \times n} \right) - \left( {6 \times \left( { - 4} \right)} \right)\]
Multiplying the terms, we get:
 \[ \Rightarrow 0 = 4n - 16 - 6n - \left( { - 24} \right)\]
 \[ \Rightarrow 0 = 4n - 16 - 6n + 24\]
Next, we need to group and combine all the like terms from the obtained equation as:
 \[ \Rightarrow 0 = \left( {4 - 6} \right)n - 16 + 24\]
Simplifying the terms, we get:
 \[ \Rightarrow 0 = - 2n + 8\]
Then, subtract 8 from each side of the equation to isolate the n term while keeping the equation balanced as:
 \[ \Rightarrow 0 - 8 = - 2n + 8 - 8\]
Simplifying the terms, we get:
 \[ \Rightarrow - 8 = - 2n + 0\]
 \[ \Rightarrow - 8 = - 2n\]
Now, divide each side of the equation by −2 to solve for n while keeping the equation balanced:
 \[ \Rightarrow \dfrac{{ - 8}}{{ - 2}} = \dfrac{{ - 2n}}{{ - 2}}\]
Evaluate the terms, we get:
 \[ \Rightarrow 4 = \dfrac{{ - 2n}}{{ - 2}}\]
 \[ \Rightarrow 4 = n\]
Hence, we get the value of n as:
 \[ \Rightarrow n = 4\]
So, the correct answer is “n = 4”.

Note: The Distributive Property is an algebraic property that is used to multiply a single value and two or more values within a set of parentheses. The distributive Property States that when a factor is multiplied by the addition of two terms, it is essential to multiply each of the two numbers by the factor, and then perform addition.
The key point to solve the given expression is that, we must know how to perform distributive property with respect to the given numbers, available in brackets that can be distributed for each number outside the bracket. We can also divide larger numbers using the distributive property by breaking those numbers into smaller factors.