Question

How do you solve: \[t - 3\left( {t + \dfrac{4}{3}} \right) = 2t + 3\]?

Answer 1

Hint: We solve the given equation for the value of unknown variable i.e. ‘t’. 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 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 \[t - 3\left( {t + \dfrac{4}{3}} \right) = 2t + 3\]...........… (1)
Use distributive property on the bracket in left hand side of the equation (1)
\[ \Rightarrow t - \left( {3 \times t} \right) - \left( {3 \times \dfrac{4}{3}} \right) = 2t + 3\]
Calculate each of the products inside each of the brackets in left hand side of the equation, cancel same terms from numerator and denominator wherever required
\[ \Rightarrow t - 3t - 4 = 2t + 3\]
Add the terms with variables together and terms as constants together on each side of the equation
\[ \Rightarrow - 2t - 4 = 2t + 3\]
Bring all constants on one side of the equation and all terms with variable on other side of the variable
\[ \Rightarrow - 4 - 3 = 2t + 2t\]
Calculate the differences on both sides of the equation
\[ \Rightarrow - 7 = 4t\]
Divide both sides of the equation by 4
\[ \Rightarrow \dfrac{{ - 7}}{4} = \dfrac{{4t}}{4}\]
Cancel same factors from numerator and denominator on both sides of the equation i.e. 4
\[ \Rightarrow \dfrac{{ - 7}}{4} = t\]
So, the value of t is \[\dfrac{{ - 7}}{4}\]

\[\therefore \]The solution of the equation \[t - 3\left( {t + \dfrac{4}{3}} \right) = 2t + 3\] is \[t = \dfrac{{ - 7}}{4}\]

Note: Students are likely to make mistake 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 take LCM of the terms inside the bracket in LHS and solve
\[ \Rightarrow t - 3\left( {\dfrac{{3t + 4}}{3}} \right) = 2t + 3\]
Cancel same factors from numerator and denominator i.e. 3
\[ \Rightarrow t - 3t - 4 = 2t + 3\]
\[ \Rightarrow - 2t - 4 = 2t + 3\]
Bring all constants on one side of the equation and all terms with variable on other side of the variable
\[ \Rightarrow - 4 - 3 = 2t + 2t\]
Calculate the differences on both sides of the equation
\[ \Rightarrow - 7 = 4t\]
Divide both sides of the equation by 4
\[ \Rightarrow \dfrac{{ - 7}}{4} = \dfrac{{4t}}{4}\]
Cancel same factors from numerator and denominator on both sides of the equation i.e. 4
\[ \Rightarrow \dfrac{{ - 7}}{4} = t\]
So, the value of t is \[\dfrac{{ - 7}}{4}\]
\[\therefore \]The solution of the equation \[t - 3\left( {t + \dfrac{4}{3}} \right) = 2t + 3\] is \[t = \dfrac{{ - 7}}{4}\]