Substitution is a mathematical technique used to simplify or solve equations by replacing a variable with a specific value. For example, if we have the equation 2x + 3 = 7 and we want to solve for x, we can use substitution by replacing x with the value that satisfies the equation, in this case x = 2. This allows us to simplify the equation and solve for the unknown variable. Substitution is a powerful tool in algebra and can be used to simplify complex equations and expressions. **

Integration by substitution is a technique used to simplify integrals by replacing a complex expression with a new variable. This new variable is chosen in such a way that it makes the integral easier to solve. The key steps in integration by substitution are to identify the inner function and its derivative, then replace the inner function with the new variable and its derivative in the integral. Finally, solve the integral with respect to the new variable and substitute back the original variable to obtain the final result. **

Back substitution is a method used to solve a system of linear equations by starting with the last equation and solving for the last variable, then using that value to solve for the second-to-last variable, and so on until all variables are determined. This method takes advantage of the fact that once a variable is solved for, its value can be substituted into the previous equations to simplify the system. By working backwards through the equations, back substitution efficiently solves for all the variables in the system. **

Substitution products in electrophilic aromatic substitution are the result of the replacement of a hydrogen atom on an aromatic ring with an electrophile. This process occurs when an electrophile attacks the aromatic ring and forms a new bond, leading to the substitution of the hydrogen atom. The resulting substitution products can be ortho, meta, or para depending on the position of the electrophile relative to the other substituents on the aromatic ring. These substitution products are important in organic chemistry as they can significantly alter the reactivity and properties of the aromatic compound. **

Integration by substitution of fractions involves rewriting a given fraction in terms of a new variable, typically denoted as u. This new variable is chosen such that it simplifies the integral and makes it easier to solve. After substituting the fraction with the new variable, the integral is then solved with respect to u. Finally, the result is converted back to the original variable to obtain the final solution. This method is particularly useful for integrating complex fractions or fractions with radicals. **

The pq-formula is used to solve quadratic equations of the form ax^2 + bx + c = 0. When using the pq-formula, the equation is first rewritten in the form (x - p)(x - q) = 0, where p and q are the solutions to the equation. To substitute the values of p and q back into the original equation, we can use the factored form to expand and simplify the equation, which will result in the original quadratic equation. This process allows us to find the solutions to the quadratic equation using the pq-formula. **

The substitution method in mathematics is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This allows us to solve for the other variable. By substituting the expression for one variable into the other equation, we can solve for the other variable and then use that value to find the value of the first variable. This method is particularly useful when dealing with linear equations and can be used to find the solution to a system of equations. **

The second substitution on aromatics involves the introduction of a second functional group onto the aromatic ring. This can be achieved through either electrophilic aromatic substitution or nucleophilic aromatic substitution. In electrophilic aromatic substitution, a second electrophile is introduced onto the aromatic ring, while in nucleophilic aromatic substitution, a second nucleophile replaces a leaving group on the aromatic ring. Both processes result in the formation of a new compound with two functional groups attached to the aromatic ring, allowing for the synthesis of a wide variety of complex organic molecules. **