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The substitution method is an algebraic technique often used to solve linear equations. It involves replacing a variable in an equation with its equivalent value to determine if the equation is true or to find the value of another variable. In the context of our exercise, you check whether a given number is a solution by substituting it for the variable in the equation.

To ensure that your use of the substitution method is successful, first identify the variable you need to replace. In our example, we substitute \(y\) with \( -2 \). After substitution, the equation should be simplified to verify if each side equals the other, confirming if the substituted value is indeed a solution. Remember, if after substitution the equation holds true (like both sides equal the same number), then the number you substituted is a solution to the equation.

Simplifying equations is a foundational skill in algebra. To simplify an equation means to make it as basic as possible while keeping it mathematically equivalent to the original. In the exercise, once we substitute \(y\) with \( -2 \), the next step is to simplify the expressions on both sides of the equation. Simplifying involves combining like terms and performing any addition or subtraction as necessary.

Combining Like Terms

Since we are dealing with simple arithmetic operations, we add or subtract the numbers to get the simplest form, which in this case will be a single number. The goal of simplification is to reach a point where the equation can be easily evaluated — ideally, to an obvious statement such as \(1=1\).

Understanding algebraic properties is crucial for manipulating and solving equations. In this case, the equation \(3+y=y+3\) illustrates a couple of important algebraic properties:

• Commutative Property of Addition: This property states that the order in which two numbers are added does not change their sum. For instance, \(3+y\) is the same as \(y+3\) because changing the order does not affect the result.
• Identity Property of Addition: This property shows that adding zero to any number does not change its value; in our equation, when substituting \(y\) with \( -2 \), we add \( -2 \) to 3 and also to the opposite side, effectively showing that the equation maintains its balance.

These properties assure us that the equation remains balanced after substitution and simplification, confirming that the proposed solution is indeed correct. Recognizing and utilizing these properties can greatly simplify the process of solving equations.