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Algebraic substitution is a fundamental technique in algebra where we replace variables with their equivalent values or expressions. In the context of our exercise, the variable 'y' is substituted with the expression it represents, which in this case is \(x - 1\). This is an essential step because it transforms the equation into terms of a single variable, making it easier to tackle.

For instance, consider our original problem \( -2x + y \) where \(y = x - 1\). By substituting \(y\) with \(x - 1\), we convert the equation into \( -2x + (x - 1) \), thereby simplifying the process to solve the equation as there is now only one variable to deal with. The power of algebraic substitution lies in its ability to clarify and streamline expressions, leading to simpler and more solvable equations.

In algebra, like terms are terms that have the exact same variable raised to the same power. They are the building blocks for simplifying algebraic expressions because they can be combined—added or subtracted. When we talk about simplifying equations, identifying like terms is a crucial step to make the expressions more manageable.

In our exercise, after substituting \(y\) with \(x - 1\), we get \( -2x + x - 1 \). Here, \( -2x \) and \(x\) are like terms because they both contain the variable \(x\) raised to the first power. They can be added together, which effectively simplifies the expression. Understanding like terms is necessary for students as it aids in solving equations more efficiently and accurately.

Simplifying equations is an essential skill in algebra that involves reducing expressions to their simplest form. It makes equations less complicated and easier to solve. This process usually includes combining like terms, as demonstrated in the previous section, and applying the distributive property to remove parentheses.

In our example, the final step of simplifying \( -2x + (x - 1) \) involves the combination of like terms. The \(x\)'s are combined to give \( -x\), and \( -1\) remains as it is because it is not a like term with any other in the equation. The simplified form of the expression is \( -x - 1\). To help students excel at simplifying equations, it's important to practice recognizing patterns within expressions and applying properties of operations consistently.