Sect. 1.3
Solving Equations
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Equivalent Equations
Addition & Multiplication Principles
Combining Like Terms
Types of Equations
But first: Awards, HW Review, and Play ?
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But, Before we go on…
Let’s play Name That Law!
a)
x+5+y = x+y+5
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b)
3a + 6 = 3(a + 2)
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c)
Associative Addition … ASSOC +
(y – 2)(3x)(y + 2) = (y – 2)(y + 2)(3x)
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f)
Reciprocals Multiplication … RECIP x
(x + 5) + y = x + (5 + y)
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e)
Distributive … DIST
7x(1 / x) = 7
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d)
Commutative Addition … COM +
Commutative Multiplication … COM x
4(a + 2b) = 8b + 4a
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COM, then DIST or DIST, then COM
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Checking for Equivalent Equations
A Solution is a Replacement Value that makes an equation True
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Two Keys for
Solving an Equation in One Variable
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We need better techniques than guessing solutions
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If we Add the same number to both sides of an
equation, it will still have the original solution
If we Multiply both sides of an equation by the same
non-0 number, it will still have the original solution
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An Example of using Horizontal Technique
for Applying the Addition Principle
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Using the Vertical Technique
to Solve the Same Equation
Compare the 2 Ways
y  4.7  13.9
 4.7  4.7
y
 18.6
y  4.7  13.9
y  4.7  4.7  13.9  4.7
y
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 18.6
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Using the Vertical Technique
can Save Steps in more complex equations
Compare the 2 Ways
7 y  11  23  6 y
7 y  11  23  6 y
 6 y  11  11  6 y
y
7 y  11  11  23  6 y  11
 34
7y
 34  6 y
7 y  6 y  34  6 y  6 y
y
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 34
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An Example of
Applying the Multiplication Principle
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Combining Like Terms
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A Term is the product of a coefficient and -1
variable(s). Examples: 9x -2x2y 11 -p
Like Terms have identical variable parts
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Combine by adding their coefficients
Example: 3xy + 11xy = (3+11)xy = 14xy
Example: -2p + 6p – p = (-2+6-1)p = 3p
Example: 4xyz + 6xy can’t be combined
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Combining a Simple Expression
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Simplify An Expression
3 x  2[4 5( x  2 y )]
 3 x  2[4  5 x  10 y ]
 3 x  8  10 x  20 y
 13 x  20 y  8
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The Opposite of an Expression
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Another “Negative” Example
9 x  5 y  (5 x  y  7)
 9 x 5 y  5x y  7
 4x  6 y  7
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Connecting Concepts:
Equations vs. Expressions
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Solving Using Both Principles
(First the Addition Principle, then Multiplication)
5 x  2( x  5)  7 x  2
5 x  2 x  10  7 x  2
3 x  10  7 x  2
 3x  2
 3x  2
12  4x
3 x
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Types of Linear Equations
Identity – Contradiction - Conditional
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We have been solving Linear Equations
A linear equation is one that can be reduced to
ax = b (a ≠ 0 and x is any variable to the 1st power)
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Don’t rush to solve them in your head
Work neatly, making each step result in an equivalent equation
Every linear equation will be in one of 3 categories:
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Conditional – it has only one value as a solution (2x = 4)
Contradiction – no value will be a solution
(x = x + 1)
Identity – every value will make the equation true (x = x)
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Solve: Is it an identity, contradiction, or a
conditional equation?
2 x  7  7( x  1)  5 x
 9t  2  2  9t  5(8  4(1  34 ))
2 x  7  7 x  7 5 x
 9t  2  2  9t  5(8  4(1  81))
2x  7  2x  7
 9t  2  2  9t  5(8  4  82)
Identity
 9t  2  2  9t  5(2  82)
3 8 x  5 7 x
 8x
 9t  2  2  9t  5 164
 8x
 9t  2  2  9t  820
 9t  2  9t  818
3  5 x
 9t
5 5
2  818
2 x
Conditional
 9t
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Contradiction
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What’s Next?
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1.4 Introduction to Problem Solving
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