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Equivalent Algebraic Equations

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Equivalent Algebraic Equations Learn and use the distributive property Rewrite equations to determine whether they are equivalent Formalize algebraic properties – PowerPoint PPT presentation

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Title: Equivalent Algebraic Equations


1
Equivalent Algebraic Equations
  • Learn and use the distributive property
  • Rewrite equations to determine whether they are
    equivalent
  • Formalize algebraic properties
  • Identify properties as they are used in solving
    equations
  • Introduce factoring as a reverse of the
    distributive property

2
  • In the previous lesson you learned to write the
    equation of line using the point-slope form. You
    were given the slope and a point.
  • But remember that a line goes through many
    points. Will the equation be equivalent if it
    written using another point?
  • In this lesson you will learn how to identify
    different equations that describe the same line.

3
  • If a line with slope 2 that passes through the
    point (-4,3) can be described by the equation y
    3 2(x4).
  • This line also passes through (1, 13), so it can
    also be described by the equation y132(x-1).

4
  • If we place both of these equations in Y1 and Y2
    in our graphing calculator, we see they produce
    the same line when graphed.
  • When a table is produced you can see that the
    same set of values is produced.
  • There are many equivalent equations that can be
    used to describe a given line.

5
The Distributive Property
  • Place the Distributive Property Template in your
    Communicator.
  • Lets picture 2(7) on grid paper.
  • One way to describe its area is to say it is
    2(43).
  • Another way is to think of it as 2(4) 2(3), by
    separating the rectangle into two parts.
  • Notice that 2(7)2(43) 2(4)2(3)14
  • This is called the distributive property.
  • Model another distributive property on the grid
    paper
  • Write the distributive property on your
    Communicator

6
The Distributive Property
  • Place the Distributive Property Template in your
    Communicator.
  • Lets picture 2(x4) on the multiplication
    rectangle.
  • Place 2 units on the left. Place x 4 across
    the top.
  • Fill in the multiplication.
  • We see that another way is to think of2(x4) is
    2(x) 2(4).
  • This is called the distributive property.
  • Model another 3(x-1) on the multiplication
    rectangle.
  • Write the distributive property on your
    Communicator

7
  • We can use the distributive property to rewrite
    some of our equations.
  • Suppose y 3 2(x 4).
  • Using the distributive property gives usy3
    2(x) 2(4) or y 3 2x 8
  • Or this can be rewritten as y 11 2x.
  • Point Slope form y 3 2(x4)
  • Slope Intercept form y 11 2x.
  • Describe what each tells us.

8
Equivalent Equations
  • Page 241

9
  • Complete steps 1-5 with your group. Be prepared
    to explain your thinking on each step.
  1. y 3 - 2(x - 1)
  2. y -5 - 2(x - 5)
  3. y 9 - 2(x 2)
  4. y 0 - 2(x - 2.5)
  5. y 7 - 2(x 1)
  6. y -9 - 2(x - 7)

10
  • Complete steps 6-7 with your group.
  1. y 2(x-2.5)
  2. y182(x-8)
  3. y52-6(x8)
  4. y-62(x4)
  5. y21-6(x4)
  6. y-14-6(x-3)
  7. y-102(x6)
  • h. 6xy 4
  • y112(x-8)
  • 12x 2y-6
  • y2(x-4)10
  • y15-2(10-x)
  • y72(x-6)
  • y-6(x0.5)
  • y-6(x2)16

11
Writing equation in different forms
  • Intercept Form y a bx
  • Point-Slope Form y y1 b(x - x1)
  • An equation of the form ax by c are said to
    be in standard form

12
Properties of Arithmetic
  • Distributive Property
  • Commutative Property of Addition
  • Commutative Property of Multiplication
  • Associative Property of Addition
  • Associative Property of Multiplication

13
Properties of Equality
  • Given that a b, for any number c
  • acbc Addition Property of Equality
  • a-cb-c Subtraction Property of Equality
  • acbc Multiplication Property of Equality
  • a/c b/c (c?0) Division Property of Equality

14
Show two equations are equivalent
  • y 2 3(x - 1)
  • y 2 2x - 3
  • y -1 3x
  • Original Equation
  • Distributive Property
  • Combine Like Terms

So y 2 3(x - 1) is equivalent to the equation
y -1 3x.
15
Show two equations are equivalent
  • 6x -2y 2
  • -2y 2 - 6x
  • y (2 - 6x)/-2
  • y -1 3x
  • Original Equation
  • Subtraction Property
  • Division Property
  • Distributive Property

So 6x 2y 2 is equivalent to the equation y
-13x.
16
Checking for Equivalency
  • You can enter the intercept form and the
    point-slope form in the calculator to verify they
    are equivalent.
  • The Standard Form (ax by c) cannot be entered
    in the calculator for verification.

17
  • By using properties of equality solve the
    equation
  • Identify the properties you use on each step.
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