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:

• Solving Systems Graphically

  • This is the graphical method.

• 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).

• Solving Systems of Linear Inequalities

  • When using the graphical method for two variables, first plot all of the lines representing the inequalities, drawing a dotted line if it is either < or >, and a solid line if it is either $\leq$ or $\geq$.
  • This is referred to as the non-graphical method.
  • Then, substitute a value for $x$ into each inequality to see that it is true.
  • The non-graphical method is much more complicated, and is perhaps much harder to visualize all the possible solutions for a system of inequalities.
  • However, when you have several equations or several variables, graphing may be the only feasible method.

• Graphing Inequalities

  • The method of graphing linear inequalities in two variables is as follows:
  • If, when substituted, the test point yields a true statement, shade the half-plane containing it.
  • If, when substituted, the test point yields a false statement, shade the half-plane on the opposite side of the boundary line.
  • Recall that, in order to graph an equation, one can substitute a value for one variable, and solve for the other.
  • Substitute $(0, 0)$ into the original inequality.

• Sequences of Mathematical Statements

  • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers.
  • For example, in the context of mathematical induction, a sequence of statements usually involves an algebraic statement into which you can substitute any natural number $(0, 1, 2, 3, ...)$ and the statement should hold true.
  • So a sequence is formed by substituting integers $k$, $k + 1 $, $k + 2$ and so on into the mathematical statement.

• Proof by Mathematical Induction

  • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (non-negative integers).
  • The inductive step: showing that if the statement holds for some $n$, then the statement also holds when $n+1$ is substituted for $n$.
  • This method works by first proving the statement is true for a starting value, and then proving that the process used to go from one value to the next is valid.
  • On the right hand side of the equation we substitute $n=0$.
  • In other words, we want to show that the statement holds true when we substitute $k+1$ for $n$:

• Binomial Expansion and Factorial Notation

  • Recall that the binomial theorem is an algebraic method of expanding a binomial that is raised to a certain power, such as $(4x+y)^7$.