Download Ch2-Section 2.2

§ 2.2
The Addition and
Multiplication Properties
of Equality
Linear Equations
Linear equations in one variable
can be written in the form ax + b = c,
where a, b and c are real numbers, and a  0.
Equivalent equations
are equations that have the same solution.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
2
Addition Property of Equality
Addition Property of Equality
If a, b, and c are real numbers, then
a = b and a + c = b + c
are equivalent equations.
Example:
a.) 8 + z = – 8
8 + (– 8) + z = – 8 + – 8
z = – 16
Add –8 to each side.
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
3
Solving Equations
Example:
4p – 11 – p = 2 + 2p – 20
3p – 11 = 2p – 18 (Simplify both sides.)
3p + (– 2p) – 11 = 2p + (– 2p) – 18
p – 11 = – 18
p – 11 + 11 = – 18 + 11
p=–7
Add –2p to both sides.
Simplify both sides.
Add 11 to both sides.
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
4
Solving Equations
Example:
5(3 + z) – (8z + 9) = – 4z
15 + 5z – 8z – 9 = – 4z
6 – 3z = – 4z
Use distributive property.
Simplify left side.
6 – 3z + 4z = – 4z + 4z Add 4z to both sides.
6+z=0
Simplify both sides.
6 + (– 6) + z = 0 +( – 6) Add –6 to both sides.
z=–6
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
5
Word Phrases as Algebraic Expressions
Example:
Write the following sentence as an equation.
The product of – 5 and – 29 gives 145.
In words: The product of
–5
and – 29
Translate:
(– 5) · (– 29)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
gives
145
=
145
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Multiplication Property of Equality
Multiplication Property of Equality
If a, b, and c are real numbers, then
a = b and ac = bc
are equivalent equations
Example:
–y=8
(– 1)(– y) = 8(– 1)
y=–8
Multiply both sides by –1.
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
7
Solving Equations
Example:
1
5
x
7
9
1  5
7 x    7
7  9
35
x
9
Multiply both sides by 7.
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
8
Solving Equations
Example:
8
x6
3
38
3
  x  6 
83
8
Multiply both sides by the reciprocal.
18 9
1
x   or 2
8 4
4
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
9
Using Both Properties
Example:
3z – 1 = 26
3z – 1 + 1 = 26 + 1
3z = 27
3 z 27

3
3
z=9
Add 1 to both sides.
Simplify both sides.
Divide both sides by 3.
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
10
Using Both Properties
Example:
12x + 30 + 8x – 6 = 10
20x + 24 = 10
Simplify the left side.
20x + 24 + (– 24) = 10 + (– 24)
Add –24 to both sides.
20x = – 14
Simplify both sides.
20 x  14

20
20
Divide both sides by 20.
7
x
10
Simplify both sides.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
11