Properties Used to Solve Equations Algebraically (Day 1) 
           Standards: REL. 1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting 
           from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.  REL.3 Solve linear 
           equations and inequalities in one variable, including equations with coefficients represented by letters. 
            
           Essential Questions: How are the properties of equality used to solve equations?  How is finding the solution to an inequality similar to finding the 
           solution to an equation?  How is solving an exponential equation different from solving a linear equation? 
            
           In algebra, when we solve equations, we use properties of equality to isolate the variable. In mathematics, it is important to follow 
           the rules when solving equations, but it is also necessary to justify, or prove that the steps we are following to solve problems are 
           correct and allowed.  
                                                                                           
                                                                        The Properties of Equality                                                                     
                              Property                                          General Rule                                     Specific Example                      
                                                                                                                                                                       
                Addition property of equality                           If a = b, then a + c = b + c.                       If x = 5, then x + 2 = 5 + 2.              
                                                                                                                                                                       
              Subtraction property of equality                          If a = b, then a – c = b – c.                      If x = 12, then x – 2 = 12 – 2              
                                                                                                                                                                       
             Multiplication property of equality                         If a = b, then a x c = b x c.                      If x = 7, then                             
                                                                                                                                                                       
                                                                                                                                                                       
                                                                                                                                                          
                 Division property of equality                    If a = b and c ≠ 0, then a ÷ c = b ÷ c.                     If         then               
                                                                                                                                                                       
                                                                                                                                                                       
             Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.                                                             
              
           Guided Practice 
           1.   Which property of equality is missing in the steps to solve the equation:  -7x + 22 = 50? 
                                       Equation                                             Steps 
                                     -7x + 22 = 50                                   Original Equation 
                                        -7x = 28                                                
                                         x = -4                                Division property of equality 
                      
           2.   Which property of equality is missing in the steps to solve the equation:            
                                                                                                              
                                       Equation                                             Steps 
                                                                                     Original Equation 
                                              
                                                                              Addition property of equality 
                                            
                                        -x = 42                                                 
                                        x = -42                                Division property of equality 
                                                  The Properties of Equality (Continued)                                                 
                             
                       Property                                General Rule                              Specific Example                
                                                                                                                                         
            Reflexive property of equality                          a = a                      (A number is always equal to itself.)     
                                                                                                                                         
                                                                                                                                         
           Symmetric property of equality                    If a = b, then b = a.                       If 5 = x, and x = 5.            
                                                                                                                                         
                                                                                                                                         
           Transitive property of equality              If a = b and b = c, then a = c.            If x = y and y = 2, then x = 2.       
                                                                                                                                         
                                                                                                                                         
          Substitution property of equality           If a = b, then b may be substituted for a in any expression containing a.          
                                                                                                                                         
          
         3. Which properties are missing in the steps to solve the equation:  82 = 5 + 7x 
          
                             Equation                              Steps 
                            82 = 5 + 7x                      Original Equation 
                              77 = 7x                                 
                               11 = x                                 
                               x = 11                                 
                                                                          
         Homework: Identify the property of equality that justifies each missing step or equation in each of the following tables. 
         1. 
                                       Equation                                                     Steps 
                 x – 1.2 = 1.9                                           Original equation 
                 x = 3.1                                                  
         2. 
                                       Equation                                                     Steps 
                 5x = 37                                                 Original equation 
                 x = 7.4                                                  
         3. 
                                       Equation                                                     Steps 
                 2x + 3 = 15                                             Original equation 
                 2x = 12                                                  
                 x = 6                                                    
          4.  
                                       Equation                                                     Steps 
                 19 = 2x – 7                                             Original equation 
                 26 = 2x                                                  
                 13 = x                                                   
                 x = 13                                                   
          
                                            Properties Used to Solve Equations Algebraically (Day 2) 
                                                                       The Properties of Operations                                           
                                Property                                        General Rule                      Specific Example            
                                                                                                                                              
                                                                                                             (      )
                     Associative property of addition                       (a + b) + c = a + (b + c)                  (   )   
                    Commutative property of addition                             a + b = b + a                                                
                                                                                                                                              
                                                                                                                                              
                  Associative property of  multiplication                   (a •b) • c = a • (b • c)            (    )
                                                                                                                          (   )               
                 Commutative property of multiplication                          a • b = b • a                                                
                                                                                                                                              
                                                                                                                                              
           Distributive property of multiplication over addition           a • (b + c) = a • b + a •  c          (     )
                                                                                                                                              
         Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number      
         system, the real number system, and the complex number system. 
         Remember: 
             When operations are performed on one side of the equation, the properties of operations are generally followed. 
             When an operation is performed on both sides of the equation, the properties of equality are generally followed. 
             If a step being taken can’t be justified, then the step shouldn’t be done. 
          
         Guided Practice 
         3.  Which properties are missing in the steps to solve the equation:  76 = 5x – 15 + 2x 
                                         Equation                                                  Steps 
                  76 = 5x – 15 + 2x                                        Original Equation 
                  76 = 5x + 2x – 15                                         
                  76 = 7x – 15                                              
                  91 = 7x                                                   
                  13 = x                                                    
                  x = 13                                                    
               
         4.   Fill in the missing properties and equation in the steps to solve the equation 5x + 3(x + 4) = 28? 
                                         Equation                                                  Steps 
                  5x + 3(x + 4) = 28                                       Original Equation 
                  5x + 3x + 12 = 28                                         
                  8x + 12 = 28                                              
                                                                           Subtraction property of equality 
                  X = 2                                                     
          
            Homework: Properties of Equality 
            Identify the property of equality that justifies each missing step or equation in each of the following tables. 
            1. 
                                                  Equation                                                                        Steps 
                      x + (x – 0.6) = 2                                                       Original equation 
                      2x – 0.6 = 2                                                             
                                                                                              Addition Property of Equality 
                      x = 1.3                                                                  
            2.          
                                                  Equation                                                                        Steps 
                      x + (4x + 32) = 12                                                      Original equation 
                      5x + 32 = 12                                                             
                      5x = –20                                                                 
                                                                                              Division property of equality 
            3.  
                                                  Equation                                                                        Steps 
                      4(x – 6) = 40                                                           Original equation 
                      x – 6 = 10                                                               
                      x = 16                                                                   
            4. 
                                                  Equation                                                                        Steps 
                      1.4 – 0.3x + 0.7x = 9.4                                                  
                      1.4 + 0.4x = 9.4                                                         
                      0.4x = 8                                                                 
                      x = 20                                                                   
             
            Solve each equation that follows. Justify each step in your process using the properties of equality. Be sure to include the properties 
            of operations, if used. 
             
            5.         7x – (4x – 39) = 0                                                6.         4(3x + 5) = –46co