Question

Solve for $a$.
\[\dfrac{{3a - 2}}{7} - \dfrac{{a - 2}}{4} = 2\]

Answer 1

Hint: In the given problem we need to solve this for ‘a’. We can solve this using the transposition method. We simplify the left-hand side of the equation by taking LCM. Then we apply mathematical operations on the like terms and the constants on the left-hand side of the equation. Then we group the ‘a’ terms on one side and constants on the other side of the equation.

Complete step by step answer:
Given, \[\dfrac{{3a - 2}}{7} - \dfrac{{a - 2}}{4} = 2\].
We know that the LCM of 7 and 4 is 28. Simplifying we have,
 \[\dfrac{{4\left( {3a - 2} \right) - 7\left( {a - 2} \right)}}{{28}} = 2\]
Now expanding the brackets, we have,
\[\dfrac{{12a - 8 - 7a + 14}}{{28}} = 2\]
\[\dfrac{{5a + 6}}{{28}} = 2\]
Cross multiplying, we have,
\[5a + 6 = 2 \times 28\]
\[5a + 6 = 56\]
We transpose ‘6’ which is present in the left hand side of the equation to the right hand side of the equation by subtracting ‘6’ on the right hand side of the equation.
\[5a = 56 - 6\]
\[5a = 50\]
Similarly, we transpose 5 to the right hand side of the equation by dividing 5 on the right hand side of the equation.
\[a = \dfrac{{50}}{5}\]
\[ \Rightarrow a = 10\]. This is the required answer.

Note:
We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘a’ in the given problem. If we get LHS = RHS then our answer is correct.
\[\dfrac{{3a - 2}}{7} - \dfrac{{a - 2}}{4} = 2\]
\[\dfrac{{3\left( {10} \right) - 2}}{7} - \dfrac{{\left( {10} \right) - 2}}{4} = 2\]
\[\dfrac{{30 - 2}}{7} - \dfrac{{10 - 2}}{4} = 2\]
\[\dfrac{{28}}{7} - \dfrac{8}{4} = 2\]
\[4 - 2 = 2\]
Simplifying we have,
\[ \Rightarrow 2 = 2\].
That is LHS=RHS. Hence the obtained is correct.
In the above, we did the transpose of addition and subtraction. Similarly, if we have multiplication, we use division to transpose. If we have division, we use multiplication to transpose. Follow the same procedure for these kinds of problems.