7 5 Solve Special Types of Linear Systems A by 35HFP4XV

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									7.5 Solve Special Types of
Linear Systems

    You will identify systems that have no solutions
    and systems that have infinitely many
    solutions.

    Essential Question: How can you identify the
    number of solutions in a linear system?
Warm-Up Exercises      Lesson 7.5, For use with pages 459-465


1. Solve the linear system.
   2x + 3y = – 9
    x – 2y = 6

ANSWER      (0, – 3)


2. You buy 8 pencils for $8 at the bookstore. Standard
   pencils cost $.85 and specialty pencils cost $1.25.
   How many specialty pencils did you buy?

ANSWER      3 specialty pencils
Independent Practice Check
 Textbook p. 447-450
Independent Practice Check
 Textbook p. 454-457
Vocabulary
       Exactly One Solution = Consistent and Independent
         Graphing – Intersecting Lines
         Substitution – Produces a Solution
         Elimination – Produces a Solution

       Infinite Solutions = Consistent and Dependent
         Graphing – Same Line
         Substitution – Variables all disappear – Left with True Sentence
         Elimination – Variables all disappear – Left with True Sentence

       No solutions = Inconsistent
        Graphing - Parallel Lines
        Substitution – Variables all disappear – Left with False
         Sentence
        Elimination – Variables all disappear – Left with False Sentence
EXAMPLE 1        A linear system with no solution

 Show that the linear system has no solution.

 3x + 2y = 10                       Equation 1

 3x + 2y = 2                        Equation 2

 SOLUTION
METHOD 1        Graphing

Graph the linear system.
EXAMPLE 1      A linear system with no solution

 ANSWER
 The lines are parallel because they have the same
 slope but different y-intercepts. Parallel lines do not
 intersect, so the system has no solution.
EXAMPLE 1     A linear system with no solution

 METHOD 2     Elimination

 Subtract the equation.
    3x + 2y = 10
    3x + 2y = 2
          0 = 8           This is a false statement.



 ANSWER
 The variables are eliminated and you are left with a
 false statement regardless of the values of x and y.
 This tells you that the system has no solution.
EXAMPLE 2      A linear system with infinitely many solutions

 Show that the linear system has infinitely many
 solutions.

     x – 2y = – 4              Equation 1

      y = 1x + 2               Equation 2
          2

 SOLUTION

 METHOD 1 Graphing

 Graph the linear system.
EXAMPLE 2     A linear system with infinitely many solutions

 ANSWER

 The equations represent the same line, so any point on
 the line is a solution. So, the linear system has
 infinitely many solutions.
EXAMPLE 2       A linear system with infinitely many solutions

 METHOD 2      Substitution

 Substitute 1 x + 2 for y in Equation 1 and solve for x.
            2
        x – 2y = – 4            Write Equation 1

 x – 2 1x + 2 = – 4             Substitute 1 x + 2 for y.
       2                                   2

       –4= –4                   Simplify.
 ANSWER

 The variables are eliminated and you are left with a
 true statement regardless of the values of x and y.
 This tells you that the system has infinitely many
 solutions.
GUIDED PRACTICE       for Examples 1 and 2

Tell whether the linear system has no solution or
infinitely many solutions. Explain.


1.     5x + 3y = 6            Equation 1
     – 5x – 3y = 3            Equation 2

METHOD 2 Elimination
Subtract the equations.
       5x + 3y = 6
     – 5x – 3y = 3
             0=9              This is a false statement.
GUIDED PRACTICE       for Examples 1 and 2


ANSWER
The variables are eliminated and you are left with a
false statement regardless of the values of x and y.
This tells you that the system has no solution.
GUIDED PRACTICE             for Examples 1 and 2


2. y = 2x – 4                      Equation 1

  – 6x + 3y = – 12                 Equation 2


METHOD 2        Substitution

Substitute 2x – 4 for y in Equation 2 and solve for x.

          – 6x + 3y = – 12          Write Equation 2

  – 6x + 3(2x – 4) = – 12          Substitute (2x – 4) for y.

                – 12 = – 12         Simplify.
GUIDED PRACTICE       for Examples 1 and 2



ANSWER
The variables are eliminated and you are left with a
true statement regardless of the values of x and y. This
tells you that the system has infinitely many solution.
EXAMPLE 3          Identify the number of solutions.

 Without solving the linear system, tell whether the
 linear system has one solution, no solution, or
 infinitely many solutions.
 a. 5x + y = – 2                  Equation 1
   –10x – 2y = 4                  Equation 2

 SOLUTION
                                  Write Equation 1 in slope-
     y = – 5x – 2                 intercept form.
                                  Write Equation 2 in slope-
    y = – 5x – 2
                                  intercept form.
 ANSWER
 Because the lines have the same slope and the same
 y-intercepts,the system has infinitely many solutions.
 They are the same line.
EXAMPLE 3       Identify the number of solutions.

 b. 6x + 2 y = 3                Equation 1
    6x + 2 y = – 5              Equation 2



 SOLUTION

 b. y = – 3x + 3        Write Equation 1 in slope- intercept form.
               2
     y = – 3x – 5       Write Equation 2 in slope-intercept form.
                2

 ANSWER
 Because the lines have the same slope but different y-
 intercepts, the system has no solution. They are parallel
 lines.
EXAMPLE 4      Write and solve a system of linear equations

 ART

 An artist wants to sell prints   Regular    Glossy    Cost
 of her paintings. She orders
                                     45        30      $465
 a set of prints for each of
 two of her paintings. Each          15        10      $155
 set contains regular prints
 and glossy prints, as shown
 in the table. Find the cost of
 one glossy print.
EXAMPLE 4      Write and solve a system of linear equations

 SOLUTION

 STEP 1

 Write a linear system. Let x be the cost (in dollars) of
 a regular print, and let y be the cost (in dollars) of a
 glossy print.

      45x + 30y = 465         Cost of prints for one painting

      15x + 10y = 155         Cost of prints for other painting
EXAMPLE 4      Write and solve a system of linear equations

 STEP 2

 Solve the linear system using elimination.

      45x + 30y = 465                    45x + 30y = 465

      15x + 10y = 155                   – 45x – 30y = – 465
                                                  0=0

 ANSWER

 There are infinitely many solutions, so you cannot
 determine the cost of one glossy print. You need more
 information.
GUIDED PRACTICE         for Examples 3 and 4

3. Without solving the linear system, tell whether it
has one solution, no solution, or infinitely many
solutions. one solution.
    x – 3y = – 15                    Equation 1
   2x – 3y = – 18                    Equation 2

SOLUTION
               x+5                 Write Equation 1 in slope-
         y=                        intercept form.
                 3
              2x + 18              Write Equation 2 in slope-
         y=                        intercept form.
                 3
ANSWER
Because the lines have different slope and different
y-intercepts,the system has one solution.
Essential Question:

How can you identify the number of
solutions in a linear system?
Independent Practice

   Textbook p. 462-465

								
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