Question

How do you solve \[78 = 6\left( { - 6a - 5} \right)\] using the distributive property?

Answer 1

Hint: Here in this question, we have to solve the given algebraic equation by using the distributive property and simplify by add or subtract the necessary term from each side of the equation to isolate the term with the variable \[a\], then multiply or divide each side of the equation by the appropriate value, while keeping the equation balanced then solve the resultant balance equation for the \[a\] value.

Complete step-by-step solution:
The Distributive Property is an algebraic property that is used to multiply a single value and two or more values within a set of parenthesis. The distributive Property States that when a factor is multiplied by the sum/addition of two terms, it is essential to multiply each of the two numbers by the factor. This property can be stated symbolically as: \[A\left( {B + C} \right) = AB + AC\]
Consider the given equation
\[78 = 6\left( { - 6a - 5} \right)\]
Use the distributive property on RHS side means multiply 6 into the each term on parenthesis i.e.,
\[ \Rightarrow \,\,\,78 = 6\left( { - 6a} \right) - 6\left( 5 \right)\]
On multiplying, we get
\[ \Rightarrow \,\,\,78 = - 36a - 30\]
Add 30 on both the sides, then
\[ \Rightarrow \,\,\,78 + 30 = - 36a - 30 + 30\]
On simplification, we get
\[ \Rightarrow \,\,\,108 = - 36a\]
To isolate the \[a\] variable by divide -36 on both the side, then
\[ \Rightarrow \,\,\,\dfrac{{108}}{{ - 36}} = \dfrac{{ - 36}}{{ - 36}}a\]
\[ \Rightarrow \,\,\, - 3 = a\]
Or, it can be rewritten as
\[ \Rightarrow \,\,\,a = - 3\]

Hence, the required solution is \[a = - 3\].

Note: The distributive property is stated as \[A\left( {B + C} \right) = AB + AC\], the equation will verify this property and using this property we have to determine the value of the variable a. We can also verify the given equation by substituting the value of “a” and while shifting the terms or variables we must take care of the sign, because while shifting it will change.