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When faced with the task of checking if a given number is a solution to an equation, one should first grasp the method of 'substitute variables'. The substitution method involves replacing the variable with the given number in the equation. For instance, if we are investigating whether (-8) is a solution for the equation (6=-3+z), we substitute (z) with (-8). This substitution gives us a new equation: 6=-3+(-8).

Through substitution, we transform an abstract equation into a concrete numerical expression that can be evaluated. This skill not only helps in solving simple equations but becomes invaluable when dealing with more complex algebraic expressions. Remember, careful substitution is key; one small mistake can lead to incorrect results.

After substituting variables, the next logical step in the problem-solving process is 'simplifying expressions'. Simplification makes it easier to see if both sides of the equation correspond. In our example, after substitution, we get the expression (6=-3+(-8)). Simplifying the right-hand side requires understanding how integers work together - in this case, combining the numbers with addition. (-3+(-8)) simplifies to (-11), leading us to a new, simplified equation: 6=-11).

Simplifying expressions is not limited to addition; it covers all arithmetic operations and the removal of parentheses, which can sometimes involve applying the distributive property or combining like terms. It is essential for revealing the underlying structure of the expression and preparing it for further analysis.

The final and crucial step in solving equations is 'verifying solutions'. After simplifying, you should have a clear statement, free from variables, to evaluate. In our case, does 6==(-11))? Clearly, these are not equivalent. Verifying solutions is more than just making sure the math adds up; it's a moment to critically assess your method and ensure no errors have occurred.

When a proposed solution does not verify, as in the equation 6=-11), we can conclude that our initial number is not a solution to the equation. This process of verification is a powerful tool for students, helping to reinforce the connection between the abstract algebraic methods and the concrete numbers that result from them. Whenever working through equations, always conclude with this step; it can save you from oversight and helps to solidify a true understanding of the mathematics involved.