substitution method

Examples of substitution method in the following topics:

• The Substitution Method

  • The substitution method is a way of solving a system of equations by expressing the equations in terms of only one variable.
  • The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation.
  • Solve this equation, and then back-substitute until the solution is found.
  • Next, we will substitute our new definition of x into the second equation:
  • Check the solution by substituting the values into one of the equations.

• Solving Systems of Equations in Three Variables

  • The substitution method of solving a system of equations in three variables involves identifying an equation that can be easily by written with a single variable as the subject (by solving the equation for that variable).
  • After that smaller system has been solved, whether by further application of the substitution method or by other methods, substitute the solutions found for the variables back into the first right-hand side expression.
  • Substitute this expression for x into the last equation in the system and solve for y:
  • Working up again, plug $(1,2)$ into the first substituted equation and solve for z:
  • Using the elimination method, begin by subtracting the first equation from the second and simplifying:

• Substitution

  • An early method of preparing phenol (the Dow process) involved the reaction of chlorobenzene with a concentrated sodium hydroxide solution at temperatures above 350 ºC.
  • To explain this, a third mechanism for nucleophilic substitution has been proposed.
  • Three additional examples of aryl halide nucleophilic substitution are presented on the right.
  • Only the 2- and 4-chloropyridine isomers undergo rapid substitution, the 3-chloro isomer is relatively unreactive.
  • Some distinguishing features of the three common nucleophilic substitution mechanisms are summarized in the following table.

• The Substitution Rule

  • Integration by substitution, also known as $u$-substitution, is a method for finding integrals.
  • When used in the latter manner, it is often known as $u$-substitution (or $w$-substitution).
  • An antiderivative for the substituted function can hopefully be determined; the original substitution between $u$ and $x$ is then undone.
  • Similar to our first example above, we can determine the following antiderivative with this method:
  • Use $u$-substitution (the substitution rule) to find the antiderivative of more complex functions

• Trigonometric Substitution

  • Trigonometric functions can be substituted for other expressions to change the form of integrands and simplify the integration.
  • Trigonometric functions can be substituted for other expressions to change the form of integrands.
  • The following are general methods of trigonometric substitution, depending on the form of the function to be integrated.
  • Note that, for a definite integral, one must figure out how the bounds of integration change due to the substitution.
  • In order to better understand these substitutions, let's go over the derivation of some of them.

• Artificial Blood Substitutes

  • The benefits to having a blood substitute are substantial.
  • As with all advances in science and medicine, the risks and benefits must be weighed for each treatment method.
  • Blood substitutes are useful for many reasons.
  • For example, the cost of blood substitutes may fall as manufacturing becomes refined.
  • There are also risks associated with blood substitutes.

• Gross Profit Method

  • The gross profit method uses the previous year's average gross profit margin to calculate the value of the inventory.
  • There are two methods to estimate inventory cost, the retail inventory method and the gross profit method.
  • Either of these methods should never be used as a substitute for performing an annual physical inventory.
  • The following is an example on how to calculate ending inventory using the gross profit method.
  • Explain how a company would use the Gross Profit Method to value inventory

• The Elimination Method

  • The elimination method is used to eliminate a variable in order to more simply solve for the remaining variable(s) in a system of equations.
  • Once the values for the remaining variables have been found successfully, they are substituted into the original equation in order to find the correct value for the other variable.
  • The elimination method can be demonstrated by using a simple example:
  • Then go back to one of the original equations and substitute the value we found for x.
  • It is always important to check the answer by substituting both of these values in for their respective variables into one of the equations.

• Other Equations in Quadratic Form

  • Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics.
  • In some special situations, however, they can be made more manageable by reducing their exponents via substitution.  
  • If a substitution can be made such that the higher order polynomial takes the form of a quadratic, any method for solving a quadratic equation can be applied.  
  • With substitution, we were able to reduce a higher order polynomial into a quadratic equation.  
  • It can now be solved with any of a number of methods (via graphing, factoring, completing the square, or by using the quadratic formula).

• Elasticity of Demand

  • So if the price of cranberry juice were to increase by $0.25 people would drink a substitute, like apple juice, instead.
  • Various research methods are used to determine price elasticity, including test markets, analysis of historical sales data, and conjoint analysis.
  • Availability of substitute goods: The more and closer the substitutes available, the higher the elasticity is likely to be, as people can easily switch from one good to another if an even minor price change is made.
  • In other words, there is a strong substitution effect.
  • If no close substitutes are available, the substitution of effect will be small and the demand inelastic.