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The substitution method is a foundational technique used to solve mathematical problems, especially algebraic expressions. Its primary purpose is to substitute quantities for variables to simplify an expression or solve equations. To effectively use this method, it's crucial to substitute the exact value for each variable.

In our exercise, the substitution method is applied by replacing each variable with its corresponding numerical value. Given that x=7 and y=5, the expression \(\frac{21}{x}+\frac{35}{y}\) is transformed by substituting 7 for x and 5 for y. This is a straightforward process but a cornerstone in solving more complex algebraic expressions and systems of equations. Proper substitution leads to a clearer and more manageable path towards finding the final solution.

Simplifying fractions is a process used to reduce fractions to their simplest form, where the numerator and denominator share no common factors other than 1. The goal of simplification is to make expressions more manageable and easier to work with. In arithmetic and algebra, simplification is not just about making numbers smaller; it also involves transforming complex expressions into simpler, equivalent forms.

In our example, the fractions \(\frac{21}{7}\) and \(\frac{35}{5}\) are reduced to \(3\) and \(7\), respectively. By dividing the numerator by the denominator, we obtain whole numbers. This step is essential because it paves the way for the final arithmetic operation that yields the solution. Learning to simplify fractions is a key skill that helps in achieving a stronger grasp of algebra and higher mathematics.

Arithmetic operations include addition, subtraction, multiplication, and division. These are the basic building blocks of mathematics that all students learn during their early years in school. Effective application of these operations is crucial when solving various mathematical problems.

In the context of our exercise, after applying the substitution method and simplifying fractions, we are left with an addition problem: adding the numbers 3 and 7. This step might seem simple, but it's a fundamental concept that leads to the final solution of the expression. The result of \(3 + 7\) is \(10\), illustrating how fundamental arithmetic operations are utilized to evaluate expressions. Grasping these operations can make the difference in understanding more complex mathematical ideas such as calculus and beyond.