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The substitution method is one of the fundamental tools in solving algebraic equations. It involves replacing variables with their respective numerical values to determine if those values satisfy the equation. This method is particularly useful when you have a single equation with one variable, as in the exercise provided.

In the given exercise, the goal is to verify whether \(m = -8\) is a solution for the equation \( -7 + m = -15 \). To apply the substitution method, you simply take the value of \(m\) and replace it with \( -8 \). After the substitution, the equation is simplified by performing the arithmetic operation left to be done, which is adding \( -7 \). By following this process, you will ascertain if the equation holds true with the given value of the variable.

Evaluating expressions is an essential skill within algebra that involves calculating the value of an expression, especially when it includes variables. This process often requires two main steps: substitution, if the value of the variable is known, followed by performing the necessary arithmetic operations such as addition, subtraction, multiplication, or division.

In our example, the expression to evaluate is \( -7 + m \), with \(m\) being the variable. The operation that needs to be performed to evaluate this expression is addition. So, after substituting \(m\) with \( -8 \), we then add \( -7 \) and \( -8 \), which results in \( -15 \). Evaluating expressions like these correctly is important, as it shows the practical use of algebra in solving real-life problems and is the first step towards solving more complex equations.

Algebraic solutions involve finding the values of variables that make an equation true. These solutions can be verified by substituting the values back into the original equation and checking if the expression evaluates to a true statement.

In relation to the exercise, after replacing \(m\) with \( -8 \) and evaluating the expression, we obtained \( -15 \) as the result, which matches the right-hand side of the equation. This confirms that the equation is satisfied, meaning \( -8 \) is indeed an algebraic solution to the equation. Understanding how to find and confirm algebraic solutions is crucial as it forms the basis for more advanced mathematics such as calculus and beyond. It's the act of unlocking the answers that algebra proposes, culminating in the rewarding assurance that the solutions provided are indeed correct.