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The substitution method is a strategy that helps to solve algebraic expressions by replacing a variable with a more complex expression. This technique is useful when working with equations that have multiple variables. By substituting, you can transform the problem into a simpler form which deals with only one variable, making it much easier to handle.

In the given exercise, you are asked to substitute the expression \(4 - 3x\) for the variable \(y\). This means every instance of \(y\) in the expressions \(5x + 2y\) and \(7x - 2y\) must be replaced with \(4 - 3x\).
- Identify where the substitution should take place, focusing on replacing \(y\) in the expression.- This conversion helps to rewrite the expression focusing only on one variable \(x\), thus simplifying the process and understanding.

Simplifying expressions in algebra involves reducing them to their simplest form. This is done by performing basic operations like addition, subtraction, distribution, and combining like terms.

In the substitution method, after you replace the variable \(y\) with \(4 - 3x\), the next step is to simplify the expression. Let’s break it down:- Distribute any coefficients by multiplying them with each term inside the parentheses. In our exercise, the coefficient 2 is distributed across the terms inside the parentheses: - Multiply 2 by each term inside: \(2 \times 4 - 2 \times 3x\) leading to \(8 - 6x\).- Next, combine any like terms. Like terms are terms that contain the same variable raised to the same power. In the expression \(5x + 8 - 6x\), you can combine \(5x\) and \(-6x\), resulting in \(-x\).- The expression finally reduces to \(-x + 8\). Each step simplifies the expression further, allowing you to clearly see its final form that focuses on a single variable.

In algebra, variables are symbols or letters, like \(x\) or \(y\), used to represent unknown values. Understanding how to manipulate these symbols is critical in solving algebraic equations and expressions. Variables enable us to express general forms of equations that can capture relationships between different quantities.

By studying how variables interact through operations, we can derive solutions to complex problems. In the context of the given exercise, variables help us perform substitutions and simplifications efficiently.
- Recognize that a letter or symbol can stand for any number. Each substitution replaces the chosen variable with another expression to help narrow down the complexities of algebra problems.- By focusing on one variable, such as \(x\), after substitution, you can oversee the effect on the overall expression and guide it towards a solution.- Mastering variable manipulation, substitution, and simplification forms a basis for further understanding of more advanced algebraic operations.