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The substitution method is a fundamental technique in algebra to find the value of variables. In this method, we replace a variable with a given number or expression, allowing us to simplify the equation and solve it more easily.

For example, if we know that the solution set for a variable is \(\{-6\}\), we can substitute this value into the equation in place of the variable and proceed with solving the equation. This helps in transforming an equation into a simpler form, which is easier to handle. Moreover, substitution is not limited to numerical values but can also involve replacing a variable with another algebraic expression, leading to a solution through further simplification and operations.

Simplifying algebraic expressions is a process of reducing complex algebraic expressions into simpler forms by performing basic arithmetic operations such as addition, subtraction, multiplication, and division.

Consider the expression \(\frac{7(-6) + 4}{b} + 13 = -6\). Simplifying inside the parenthesis first and then performing division by \(b\), the expression becomes \(\frac{-42 + 4}{b} + 13\). Further simplification by combining like terms leads to \(\frac{-38}{b} + 13\), which is much easier to solve. The skills of simplification are crucial to handling equations with fractions and isolating the variables effectively.

The goal of isolating variables is to rewrite an equation so that one variable stands alone on one side of the equation. This is achieved by performing a series of algebraic operations that reverse the operations surrounding the variable.

To isolate \(b\) in the simplified equation \(\frac{-38}{b} + 13 = -6\), we start by eliminating other terms (such as 13) from the side of the equation with \(b\). Then, we can manipulate the equation so \(b\) is the only term on one side. In this case, by multiplying both sides by \(b\), we remove \(b\) from the denominator, and then we divide by the coefficient of \(b\) (-19) to get \(b = 2\). Thus, the variable \(b\) is successfully isolated, furnishing us with the solution.