Question

Check the distributive property of multiplication over addition if $a = \dfrac{{ - 1}}{2},b = \dfrac{4}{3},c = \dfrac{{ - 2}}{7}$ . Then $\left[ {ax\left( {b + c} \right) = a \times b + a \times c} \right]$ .

Answer 1

Hint: So for checking whether this follows the distributive property of multiplication over addition or not we have to substitute the values in the given equation and then we have to solve for them. If they are equal then yes it follows and if they not then we can say that it does not follow.

Complete step-by-step answer:
we have the equation given as $\left[ {a \times \left( {b + c} \right) = a \times b + a \times c} \right]$ and values are given by $a = \dfrac{{ - 1}}{2},b = \dfrac{4}{3},c = \dfrac{{ - 2}}{7}$ . So on taking the LHS and RHS one by one, we get
So, the LHS will be
$ \Rightarrow a \times \left( {b + c} \right)$
On substituting the values, we will get the equation as
$ \Rightarrow \dfrac{{ - 1}}{2} \times \left( {\dfrac{4}{3} + \left( {\dfrac{{ - 2}}{7}} \right)} \right)$
On solving the innermost braces, we get
$ \Rightarrow \dfrac{{ - 1}}{2} \times \left( {\dfrac{4}{3} - \dfrac{2}{7}} \right)$
Now on taking the LCM and solving it we get the equation as
$ \Rightarrow \dfrac{{ - 1}}{2} \times \left( {\dfrac{{28 - 6}}{{21}}} \right)$
And on solving it, we will have
$ \Rightarrow \dfrac{{ - 1}}{2} \times \left( {\dfrac{{22}}{{21}}} \right)$
And on multiplying the above equation, we get
$ \Rightarrow \dfrac{{ - 11}}{{21}}$
Now we will solve the RHS by distributive law,
$ \Rightarrow \left[ {a \times \left( {b + c} \right) = a \times b + a \times c} \right]$
So by taking the RHS and substituting the values, we get
$ \Rightarrow \dfrac{{ - 1}}{2} \times \dfrac{4}{3} + \dfrac{{ - 1}}{2} \times \dfrac{{ - 2}}{7}$
And on solving the multiplication in the above line, we get
$ \Rightarrow \dfrac{{ - 2}}{3} + \dfrac{1}{7}$
On adding it and solving it, we get
$ \Rightarrow \dfrac{{ - 11}}{{21}}$
Here, we can see that the LHS is equal to the RHS.
Hence, the distributive property of multiplication over addition is verified.

Note: So this property can also be used when we have to multiply or divide the expressions which are algebraic and it includes the variables and also the real numbers. And it will be said to be as the distributive property having variables. Also, we should know that the expression can be anything, it can be monomial or polynomial, or also it can be binomial.