Question

How do you solve \[ - 14 = \dfrac{2}{5}(9 - 2b)\] ?

Answer 1

Hint: We solve the given equation for the value of an unknown variable i.e. ‘b’. Cross multiply the denominator from one side to opposite side. Multiply the values outside the bracket to the terms inside the bracket using distributive law of multiplication over addition. Write equations on both sides in simplest form. Shift all variable values on one side and constant values on the other side of the equation. Calculate the value of the variable by dividing with the suitable value.
* Distributive Property: For any three numbers ‘a’, ‘b’ and ‘c’ we can write \[a(b + c) = ab + bc\]

Complete step by step solution:
We are given the equation \[ - 14 = \dfrac{2}{5}(9 - 2b)\] … (1)
Cross multiply the denominator from right hand side to left hand side of the equation
\[ \Rightarrow - 14 \times 5 = 2(9 - 2b)\]
Calculate the product on left hand side of the equation
\[ \Rightarrow - 70 = 2(9 - 2b)\]
Use distributive property on the bracket in right hand side of the equation
\[ \Rightarrow - 70 = (2 \times 9) - (2 \times 2b)\]
Calculate each of the products inside each of the brackets in right hand side of the equation
\[ \Rightarrow - 70 = 18 - 4b\]
Bring all constants on one side of the equation and all terms with variable on other side of the variable
\[ \Rightarrow - 70 - 18 = - 4b\]
Calculate the values on both sides of the equation
\[ \Rightarrow - 88 = - 4b\]
Divide both sides of the equation by -4
\[ \Rightarrow \dfrac{{ - 88}}{{ - 4}} = \dfrac{{ - 4b}}{{ - 4}}\]
Cancel same factors from numerator and denominator on both sides of the equation i.e. 4
\[ \Rightarrow 22 = b\]
So, the value of b is 22
\[\therefore \]The solution of the equation \[ - 14 = \dfrac{2}{5}(9 - 2b)\] is \[b = 22\]

Note:
Students are likely to make mistakes while shifting the values from one side of the equation to another side of the equation as they forget to change the sign of the value shifted. Keep in mind we always change the sign of the value from positive to negative and vice versa when shifting values from one side of the equation to another side of the equation.
Alternate method:
We can directly multiply the value outside the bracket to each term inside the bracket
\[ \Rightarrow - 14 = \left( {\dfrac{2}{5} \times 9} \right) - \left( {\dfrac{2}{5} \times 2b} \right)\]
Calculate each product inside the brackets
\[ \Rightarrow - 14 = \dfrac{{18}}{5} - \dfrac{{4b}}{5}\]
Shift constant values to left hand side of the equation
\[ \Rightarrow - 14 - \dfrac{{18}}{5} = \dfrac{{4b}}{5}\]
Take LCM in left hand side of the equation
\[ \Rightarrow \dfrac{{ - 70 - 18}}{5} = \dfrac{{4b}}{5}\]
\[ \Rightarrow \dfrac{{ - 88}}{5} = \dfrac{{4b}}{5}\]
Cancel same terms from both sides of the equation
\[ \Rightarrow 22 = b\]
\[\therefore \] The solution of the equation \[ - 14 = \dfrac{2}{5}(9 - 2b)\] is \[b = 22\]