Transcript 3.2:Solving Equations

3.2
Solving Equations:
Review of the Addition
and Multiplication
Properties
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Equation vs. Expression
Statements like 5 + 2 = 7 are called equations.
An equation is of the form expression = expression.
An equation can be labeled as
Equal sign
x + 5 = 9
left side
right side
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Addition Property of Equality
Let a, b, and c represent numbers.
If a = b, then
a+c=b+c
and
a–c=b-c
In other words, the same number may be added
to or subtracted from both sides of an equation
without changing the solution of the equation.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Multiplication Property of Equality
Let a, b, and c represent numbers and let c  0.
If a = b, then
a b
a ∙ c = b ∙ c and c = c
In other words, both sides of an equation may
be multiplied or divided by the same nonzero
number without changing the solution of the
equation.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Solve for x.
x-4=3
To solve the equation for x, we need to rewrite the
equation in the form
x = number.
To do so, we add 4 to both sides of the equation.
x-4=3
x - 4 + 4 = 3 + 4 Add 4 to both sides.
x=7
Simplify.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Check
To check, replace x with 7 in the original equation.
x - 4 = 3 Original equation
?
7 - 4 = 3 Replace x with 7.
3 = 3 True.
Since 3 = 3 is a true statement, 7 is the solution
of the equation.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Solve for x
4x = 8
To solve the equation for x, notice that 4 is
multiplied by x.
To get x alone, we divide both sides of the equation
by 4 and then simplify.
4x 8
=
4
4
1∙x = 2 or x = 2
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Check
To check, replace x with 2 in the original
equation.
4x = 8 Original equation
4 ∙ 2 = 8 Let x = 2.
?
8 = 8 True.
The solution is 2.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
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Using Both Properties to Solve Equations
2(2x – 3) = 10
Use the distributive property to simplify the left side.
4x – 6 = 10
Add 6 to both sides of the equation
4x – 6 + 6 = 10 + 6
4x = 16
Divide both sides by 4.
x=4
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
9
Check
To check, replace x with 4 in the original equation.
2(2x – 3) = 10
2(2 · 4 – 3) = 10
?
2(8 – 3) = 10
?
(2)5 = 10
Original equation
Let x = 4.
True.
The solution is 4.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra, 6ed
10