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Substitution is a fundamental concept in algebra that involves replacing variables with their given values to simplify an expression or solve an equation. When dealing with expressions like \(5x + x\), the process of substitution helps us evaluate the expression for a specific value of \(x\).

To effectively use substitution, follow these steps: First, identify the variable in the expression and the value it should be replaced with. In our example, the variable is \(x\), and we are given that \(x=7\). Next, replace every instance of the variable in the expression with the given value. So, \(5x + x\) becomes \(5(7) + 7\). This transformed expression now contains no variables, just numbers and operations, and can be easily evaluated using arithmetic. Substitution serves as the bridge between abstract algebraic expressions and concrete numerical operations.

Simplifying expressions is the process of combining like terms and performing arithmetic operations to condense an algebraic expression into its simplest form. For the expression \(5x + x\), simplification starts with recognizing that both terms contain the common variable \(x\), making them 'like terms'.

In simplifying the given expression, you would combine the coefficients of like terms, which are the numerical factors in front of variables. Add them together to get \(5 + 1\) times \(x\), which simplifies to \(6x\). Once substitution has occurred, in this case \(x = 7\), you would multiply \(6\) by \(7\) to get \(42\), the simplest form of the original expression when \(x = 7\). Simplifying expressions reduces complexity and clarifies the core components of an algebraic expression.

Algebraic operations, including addition, subtraction, multiplication, and division, are the building blocks of algebra and are used to manipulate algebraic expressions. When evaluating \(5x + x\), we use multiplication to combine the coefficient 5 with the variable x, and addition to combine the terms \(5x\) and \(x\).

After substitution, the expression \(5(7) + 7\) involves executing the multiplication operation first, due to the order of operations (often remembered by the acronym PEMDAS—which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Once we have calculated \(5(7)\) to get \(35\), we proceed with the addition operation to add \(7\), simplifying to \(42\). It is essential to understand and correctly apply algebraic operations when evaluating expressions to ensure accurate and meaningful conclusions in algebraic problem-solving.