Readings for Session 11 – (Continued)  

Solving Equations

Solution to an Equation:   Remember that a value is a solution to an equation when that value is substituted in for the variable in the equation, the resulting arithmetic statement is a true statement.

Example:  Is 6 a solution to the equation 42 ÷ x = 7? 

Yes, since 42 ÷ 6 = 7 is a true statement, 6 is a solution to this equation. 

We state that 6 is a solution to this equation by writing x = 6.

We write the solution set as {6}.

Example:  The value 4 is not a solution to the equation 10 ÷ x = 5 since 10 ÷ 4 = 2 R.2 ≠ 5.

Example:  The value 12 is a solution to 5x = 60 since 5 ∙ 12 = 60.

The solution set for the equation is {12}.

 Multiplication Property of Equality:  If we multiply both sides of an equation by the same non-zero value, the resulting statement is still an equation and it has the same solution set as the original equation.  

General Property:  If a = b and c ≠ 0, then ac = bc.

Note what happens when we use this property with the missing-factor definition for division.

      a ÷ b = c

b(a ÷ b) = bc           Multiplication Property of Equality

                = a              Since b ∙ c = a.

     The last step uses the fact that a ÷ b = c  means b ∙ c = a and substitution. This gives a useful cancellation property for solving equations.

Cancellation Properties: The Cancellation Property for Multiplication and Division of Whole Numbers says that if a value is multiplied and divided by the same nonzero number, the result is the original value.

General Properties:  b(a ÷ b) = a and (a ∙ b) ÷ b = a

          Note that in the following example, we use the algebraic form of the division sign. Also, note the first step uses the Multiplication Property of Equality and the second step uses a cancellation property.

Example:

 

Division Property of Equality:  If we divide both sides of an equation by the same non-zero value, the resulting statement is still an equation and it has the same solution set as the original equation.

General Property:  If a = b and c ≠ 0, then a ÷ c = b ÷ c.

Challenge Problem: Can you show the second cancellation property? That is show that (a ∙ b) ÷ b = a. 

Once again, we use the algebraic form of the division sign and a cancellation property.

Example:

 

    

      

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