Algebra II

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Algebra II

• Section 1-4
• Solving Equations

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What You'll LearnWhy It's Important

• To translate verbal expressions and sentences
  into algebraic expressions and equations
• To solve equations by using the properties of
  equality, and
• To solve equations for a specific variable
• You can use equations to solve problems involving
  history, astronomy, and travel.

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Word Expressions?Algebraic Expressions

• The language of today's algebra provides a
  powerful way to translate word expressions into
  algebraic or mathematical expressions.
• Variables are used to represent numbers that
  aren't known
• Any letter can be used as a variable.

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Word Expressions?Algebraic Expressions

• The language of today's algebra provides a
  powerful way to translate word expressions into
  algebraic or mathematical expressions.
• Variables are used to represent numbers that
  aren't known
• Any letter can be used as a variable.

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Example 1Fill in the boxes with other words that
mean the same thing
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Example 1Fill in the boxes with other words that
mean the same thing
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Equations

• Sentences with variables to be replaced,
• such as 4x 8 32 and 2x 4 gt 9 are called
  open sentences
• An open sentence that states that two
  mathematical expressions are equal is called an
  equation
• Equations can be used to represent verbal
  mathematical sentences

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Equations

• Sentences with variables to be replaced,
• such as 4x 8 32 and 2x 4 gt 9 are called
  open sentences
• An open sentence that states that two
  mathematical expressions are equal is called an
  equation
• Equations can be used to represent verbal
  mathematical sentences

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How to Solve an Equation

• To solve an equation, you find replacements for
  the variable that make the equation true
• Each of these replacements is called a solution
  of the equation
• We can use certain properties of equality to
  solve equations or open sentences

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Example 2

• Sometimes an equation can be solved by adding or
  subtracting the same number on each side.
• Solve x 54.57 78

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Example 2 Solution

• Sometimes an equation can be solved by adding or
  subtracting the same number on each side.
• Solve x 54.57 78
• - 54.57 - 54.57
• x 23.43
• Check
• 23.43 54.57 ? 78
• 78 78 ?

Subtract 54.57 from each side
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Example 3a

• Some equations may be solved by multiplying or
  dividing each side by the same number
• Solve 4x -12

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Example 3a

• Some equations may be solved by multiplying or
  dividing each side by the same number
• Solve 4x -12
• x -3
• Check
• 4(-3) ? -12
• -12 -12 ?

Divide both sides by 4
This equation could also be solved by multiplying
each side by ¼, the reciprocal of 4.
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Example 3b

• Solve

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Example 3b

• Solve
• t -20
• Check
• -¾(-20)?15
• 1515?

Divide each side by -¾ which is the same as
Multiply each side by , the reciprocal of
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Example 4

• In order to solve some equations, it may be
  necessary to apply more than one property.
• Solve 3(2a 25) - 2(a - 1) 78

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Example 4

• In order to solve some equations, it may be
  necessary to apply more than one property.
• Solve 3(2a 25) - 2(a - 1) 78
• 6a 75 - 2a 2 78
• 4a 77 78
• -77 -77
• 4a 1
• a ¼

distributive substitution properties
commutative, distributive substitution
properties
subtraction substitution properties
division substitution properties
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Example 4 Check

• Solve 3(2a 25) - 2(a - 1) 78
• 3(2¼ 25) - 2(¼ - 1) 78
• 3(25½) 2(-¾) ? 78
• 76½ 1½ ? 78
• 78 78 ?

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Example 5Integration Geometry

• Sometimes you need to solve an equation or
  formula for a variable.
• The formula for the volume of a cylinder is V
  ?r2h, where V is the volume, r represents the
  radius of the circular base and top and h
  represents the height of the cylinder. Solve the
  formula for h.

r
h
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Example 5Integration Geometry

• The formula for the volume of a cylinder is V
  ?r2h, where V is the volume, r represents the
  radius of the circular base and top and h
  represents the height of the cylinder. Solve the
  formula for h.
• V ?r2h

r
Divide each side by ?r2
h
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Problem Solving Plan

• You can use a four-step problem-solving plan to
  help you solve problems
• 1. Explore the problem
• 2. Plan the solution
• 3. Solve the problem
• 4. Examine the solution

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Example 6Integration Geometry

• The perimeter of a parallelogram is 48 inches.
  What is the length of the longer side if the
  shorter side measures 9 inches?

l
9 in.
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Example 6 SolutionIntegration Geometry
l
9 in.

• The perimeter of a parallelogram is 48 inches.
  What is the length of the longer side if the
  shorter side measures 9 inches?
• Explore Draw a diagram and let l represent the
  measure of the longer side
• Plan The perimeter equals the sum of the lengths
  of the sides. So, we can write the following
  equation
• 2(9) 2(l) 48
• Solve 18 2l 48
• -18 -18
• 2l 30
• l 15 The length of
  the longer side is 15 inches
• Examine If one of the two longer sides has
  length 15 inches and one of the shorter sides has
  length 9 inches, the perimeter is 15 15 9
  9 48 inches. Thus, the answer is correct.

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THE END