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Understanding how to combine like terms is essential when solving linear equations like the example given, where we have the equation 2m + 5m = 49. Like terms are terms that have the same variable raised to the same power. In this case, both terms have the variable m and since they are both to the first power, we can add them together as if they were regular numbers.

To combine the like terms, we simply add the numerical coefficients. Think of it like apples: If you have 2 apples (2m) and someone gives you 5 more apples (5m), you now have 7 apples (7m). Thus, by combining 2m and 5m, we get 7m. This simplifies the equation, making it easier to solve by concentrating the variable m into a single term on one side of the equation.

Once we've combined like terms, the next step in solving linear equations is isolating the variable. This means we want to get the variable m by itself on one side of the equation. In the exercise, we've combined the terms to get 7m = 49. To isolate m, we need to undo the multiplication of m by 7. This is achieved by doing the inverse operation, which is division in this case.

We divide both sides of the equation by 7 to maintain equality (what you do to one side, you must do to the other). Thus, m is isolated when we calculate m = 49/7, giving us m = 7. It is crucial to perform the same operation on both sides to keep the equation balanced, as this is the foundation of algebra.

After finding the solution m = 7, it's important to check that it satisfies the original equation. This step is like quality control to ensure we haven't made a mistake. To check our solution, we plug 7 back into the original equation 2m + 5m = 49 in place of m.

By substituting m with 7, we get 2(7) + 5(7), which simplifies to 14 + 35, equaling 49. Since the left-hand side of the equation matches the right-hand side, our solution m = 7 is correct. It is a crucial step to ensure accuracy and a good habit to develop for more complex problems, where it is easy to make mistakes.