Solve the following equation and check your results.
\[\dfrac{{\left( {2m + 5} \right)}}{3} = \left( {3m - 10} \right)\]

Hint: Here, we need to solve the equation and check our results. We will use the operations of addition, subtraction, multiplication, and division to find the value of \[m\]. Then, we will substitute the value of \[m\] in the given equation to check our results.

Complete step-by-step answer:
We will First use the basic mathematical operations i.e. addition, subtraction, multiplication, and division to find the value of \[m\].
Multiplying both sides of the equation by 3, we get
\[\begin{array}{l} \Rightarrow \dfrac{{\left( {2m + 5} \right)}}{3} \times 3 = \left( {3m - 10} \right) \times 3\\ \Rightarrow 2m + 5 = 3\left( {3m - 10} \right)\end{array}\]
Multiplying 3 by \[\left( {3m - 10} \right)\] using the distributive law of multiplication, we get
\[\begin{array}{l} \Rightarrow 2m + 5 = 3 \cdot 3m - 3 \cdot 10\\ \Rightarrow 2m + 5 = 9m - 30\end{array}\]
Adding 30 on both sides, we get
\[\begin{array}{l} \Rightarrow 2m + 5 + 30 = 9m - 30 + 30\\ \Rightarrow 2m + 35 = 9m\end{array}\]
Subtracting \[2m\] from both sides, we get
\[\begin{array}{l} \Rightarrow 2m + 35 - 2m = 9m - 2m\\ \Rightarrow 35 = 7m\end{array}\]
Finally, dividing both sides of the equation by 7, we get
\[ \Rightarrow \dfrac{{35}}{7} = \dfrac{{7m}}{7}\]
Therefore, we get
\[ \Rightarrow 5 = m\]
Thus, we get the value of \[m\] as 5.
Now, we will check our answer by using the given equation.
If left hand side is equal to right hand side, then our answer is correct.
Substituting \[m = 5\] in the left hand side (L.H.S.) of the given equation \[\dfrac{{\left( {2m + 5} \right)}}{3} = \left( {3m - 10} \right)\], we get
\[ \Rightarrow L.H.S. = \dfrac{{\left( {2 \times 5 + 5} \right)}}{3}\]
Multiplying 2 by 5, we get
\[ \Rightarrow L.H.S. = \dfrac{{\left( {10 + 5} \right)}}{3}\]
Adding 10 and 5, we get
\[ \Rightarrow L.H.S. = \dfrac{{15}}{3}\]
Dividing 15 by 3, we get
\[ \Rightarrow L.H.S. = 5\]
Substituting \[m = 5\] in the right hand side (L.H.S.) of the given equation \[\dfrac{{\left( {2m + 5} \right)}}{3} = \left( {3m - 10} \right)\], we get
\[ \Rightarrow R.H.S. = \left( {3 \times 5 - 10} \right)\]
Multiplying 3 by 5, we get
\[ \Rightarrow R.H.S. = \left( {15 - 10} \right)\]
Subtracting 10 from 15, we get
\[ \Rightarrow R.H.S. = 5\]
Therefore, we can observe that
\[ \Rightarrow L.H.S. = R.H.S.\]
Thus, the value \[m = 5\] satisfies the given equation.
Hence, we have verified our answer.

Note: The given equation is a linear equation in one variable in terms of \[m\]. A linear equation in one variable is an equation that can be written in the form \[ax + b = 0\], where \[a\] is not equal to 0, and \[a\] and \[b\] are real numbers. For example, \[x - 100 = 0\] and \[100P - 566 = 0\] are linear equations in one variable \[x\] and \[P\] respectively. A linear equation in one variable has only one solution.
We have used the distributive law of multiplication in the solution to multiply 3 by \[\left( {3m - 10} \right)\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].