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In algebra, the substitution method is a crucial technique for evaluating algebraic expressions. It involves replacing the variables in an expression with their given values. By doing this, you simplify the expression to a number.

For example, in the equation \(4x + 2y + 7\), you can substitute specific values for \(x\) and \(y\). If we substitute \(x = 2\) and \(y = 1\), the expression becomes: \[4 * 2 + 2 * 1 + 7\]

This method allows you to break down and solve complex expressions step by step, making it easier to manage.

Basic algebra forms the foundation of solving more advanced mathematical problems. It involves working with variables, constants, and arithmetic operations. When evaluating algebraic expressions, you often deal with adding, subtracting, multiplying, or dividing terms like \(4x\) or \(2y\).

In the given problem, the expression \(4x + 2y + 7\) comprises terms involving variables \(x\) and \(y\), along with a constant 7. This is a linear combination of variables and constants.

Understanding these basic principles is essential for solving and simplifying algebraic expressions efficiently and accurately.

A step-by-step approach in solving algebraic expressions makes the problem more manageable and ensures that no details are missed. Let's go through the steps for the provided exercise.

First, identify the expression: \[4x + 2y + 7\]. We need to evaluate this for two sets of values.

For \(x = 2\) and \(y = 1\):
Substitute the values: \[4 * 2 + 2 * 1 + 7\]
1. Multiply: \(4 * 2 = 8\)
2. Multiply: \(2 * 1 = 2\)
Add the results: \[8 + 2 + 7 = 17\]

For \(x = 1\) and \(y = 5\):
Substitute the values: \[4 * 1 + 2 * 5 + 7\]
1. Multiply: \(4 * 1 = 4\)
2. Multiply: \(2 * 5 = 10\)
Add the results: \[4 + 10 + 7 = 21\]

This methodical progression ensures each mathematical operation is clear and correct, leading to accurate outcomes.