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Understanding how to substitute values into an equation is fundamental for solving linear equations. When working with an equation like \(\frac{7x+4}{b}+13=x\), the goal is to find the unknown variable value that makes the equation true. In our exercise, the solution set is given as \(\{-6\}\), which means the value of \(x\) is \( -6\). Substituting this into the equation transforms it from an abstract formula to a specific, solvable expression.

To substitute the value correctly, simply replace every occurrence of \(x\) with \( -6\), ensuring that you follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this instance, the equation simplifies to \(\frac{-38}{b}+13=-6\), setting the stage for solving the equation and finding the value of \(\b\).

Simplifying equations is a process of transforming a complicated expression into a simpler one that is easier to solve. It often involves combining like terms, distributing multiplication over addition or subtraction, and moving terms from one side of the equation to the other to isolate the variable. When given the expression \(\frac{-38}{b}+13=-6\), the goal is to simplify it such that \(b\) is by itself on one side of the equation.

To do this, we perform a series of inverse operations. We want to undo the addition of 13 to isolate the fraction \(\frac{-38}{b}\). By subtracting 13 from both sides, we get \(\frac{-38}{b}=-19\), which is a much simpler expression. This step is vital as it bridges the transition from a complex equation to one that is straightforward and nearing its solution. Through simplification, we can then cross-multiply to solve for \(b\), leading us closer to the sought-after answer.

A solution set is a collection of all possible values that satisfy a given equation. In the context of linear equations, where there is just one variable, the solution set usually contains a single number. However, it's possible for an equation to have no solutions (inconsistent), one solution (consistent), or infinitely many solutions (dependent).

For the equation \(\frac{7x+4}{b}+13=x\), with the solution set \(\{-6\}\), there is exactly one value of \(x\) that satisfies the equation. Once we simplify the equation, as in our example, it becomes clear that when \(b=2\), the equality holds true for \(x=-6\). The value \(b=2\) is then the particular solution that forms our solution set for \(b\), which in this case is also a single number. It's crucial to note that solution sets can look different depending on the equation, such as a list of numbers, range of numbers, or even algebraic expressions.