Chapter 7 Lesson 2 Solving Equations with Grouping Symbols pgs

							          Chapter 7 Lesson 2
         Solving Equations with
          Grouping Symbols
              pgs. 334-338
           What you will learn:
 Solve equations that involve grouping
                 symbols
Identify equations that have no solution
   or an infinite number of solutions
           Vocabulary

• Null/empty set (336): equations that
     have no solution. No value of the
     variable results in a true sentence.
     Represented by  or { }
• Identity (336): an equation that is true
     for every value of the variable
 Josh starts walking at a rate of 2 mph.
One hour later, his sister Maria starts on
  the same path on her bike, riding at
                 10mph

                Rate         Time       Distance
               (mph)        (hours)      (miles)
    Josh         2             t           2t
   Maria         10            t-1        10(t-1)
  What does t represent?     The time Josh travels

  Why is Maria’s time shown as t-1?
                    She left 1 hour later than Josh
  Write an equation that represents the time
    when Maria catches up to Josh.      2t = 10(t-1)
Example 1: Solve Equations
    with Parentheses
Solve the equation from the previous
 chart: 2t = 10(t-1)
 Write the problem: 2t = 10(t -1)
Distributive Property: 2t = 10(t) - 10(1)
                Simplify: 2t = 10t -10
Subtract 10t from each side: 2t -10t = 10t - 10t -10
                   Simplify: -8t = -10
      Divide each side by -8: -8t = -10
                              -8     -8
                   Simplify/Solve: t = 5 or 1 1/4
                                        4
     Now check the previous problem.


Josh traveled 2 miles  5 hour or 2.5 miles
                hour       4
Maria traveled 1 hour less than Josh. She
 traveled: 10 miles  1 hour
              hour       4       or 2.5 miles
 Therefore, Maria caught up to Josh in 1/4
 hour or 15 minutes.
          Another Example 1: Solve
         Equations with Parentheses
     Solve: 6(n-3) = 4(n + 2.1)
Distributive Property on both sides: 6(n) - 6(3) = 4(n) + 4(2.1)
                           Simplify: 6n - 18 = 4n + 8.4
        Subtract 4n from each side: 6n -4n -18 = 4n -4n + 8.4
                           Simplify: 2n - 18 = 8.4
               Add 18 to both sides: 2n - 18 +18 = 8.4 + 18
                            Simplify: 2n = 26.4
                                                    Simplify/Solve:
             Divide both sides by 2: 2n = 26.4         n = 13.2
                                     2     2
                                              Check your solution!
      Example 2: Use an Equation to
           Solve a Problem
   The perimeter of a rectangle is 20 feet. The
     width is 4 feet less than the length. Find the
     dimensions of the rectangle. Then find its
     area.
Words: The width is 4 feet less than the length. The perimeter
           is 20 feet.
   Symbols: Let A = area
             Let L-4 = width
        2length + 2width = perimeter
            Equation: 2length + 2(L-4) = 20
                      2L + 2L - 8 = 20
                      4L - 8 = 20
                      4L = 28
                       L=7
• Since we know the length is 7 ft, now
  we need to find the width.
   Formula: 2L + 2W = Perimeter
            2(7) + 2W = 20
            14 + 2W = 20
                 2W = 6
                   W = 3 So the width is 3 feet

  Check: 2(7) + 2(3) = 20           Now find the area
          14 + 6 = 20               Of the rectangle.
           20 = 20 
                                    A = LW
                                    A = 73
                                    A = 21 ft2
  Example 3: No Solution

Solve: 12 - h = -h + 3
Add an h to both sides: 12 - h + h = -h + h + 3
              Simplify: 12 = 3

      The sentence 12 = 3 is never true. So the
      Solution set is 
   Example 4: All Numbers as
             solutions
Remember, an equation that is true
 for every value of the variable is
        called an identity.
Solve: 3(2g + 4) = 6(g+2)
      Distributive Property: 3(2g) + 3(4) = 6(g) + 6(2)
                  Simplify: 6g + 12 = 6g + 12

Subtract 12 from each side: 6g + 12 - 12 = 6g + 12 - 12
                  Simplify: 6g = 6g

     Mentally divide each side by 6: g = g
     The sentence g = g is always true, the solution set
     is all numbers.
                Your Turn!
     Solve each equation. Check your
                 solution
                      a = 11Check: 3(11-5) = 18
A. 3(a-5) = 18                         3(6) = 18
                                         18 = 18 
                     S = 18 Check: 3(18+22) = 4(18+12)
B. 3(s+22) = 4(s+12)                 3(40) = 4(30)
                                       120 = 120
C. 4(f+3) + 5 = 17 + 4f
                     f=f   The solution set is all numbers
D. 8y - 3 = 5(y - 1) +3y
                     The solution set is 
            One More!

Find the dimension of the rectangle.
P = 460ft                2(w) + 2(w+30)=460
                             2w + 2w + 60 = 460
                             4w + 60 = 460
                           w 4w + 60-60 = 460 -60
                             4w = 400
                             w =100
           w + 30
                               w + 30 =Length
                               100 + 30 = Length
                               130 = L
          So the the rectangle is 130ft by 100ft
• PRACTICE IS BY THE DOOR ON
  YOUR WAY OUT!

• QUIZ TOMORROW OVER 7-1 & 7-2