Download Equations The Addition Property of Equality

Math 90
2.1 "Solving Basic Equations"
Objectives:
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Learn about equations.
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The addition, subtraction, multiplication, and division property of equality.
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Markdown and markup.
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Percent.
Equations
De…nition:
"Equation"
An equation is a number sentence that says that the expressions on either side of the equals sign, = ,
represent the same number. Examples:
x + 5; 2x
5 = 11; and 3x2
4x + 5 = 0
:
Solution of an Equation:
Any replacement for the variable that makes an equation true is called a solution of the equation.
To solve an equation means to …nd all of its solutions.
One way to determine whether a number is a solution of an equation is to evaluate the expression on each side of the
equals sign by substitution. If the values are the same, then the number is a solution.
Example 1: (Checking for solutions of an equation)
Determine whether the given number is a solution of the given equation.
y
a) 15; x + 17 = 32
b) 49;
=6
8
c)
4; 3 (y
1) =
15
The Addition Property of Equality
De…nition:
"Equivalent Equations"
kTwo equations are equivalent equations when they have the same solutions.k
The Addition Property of Equality:
For any real numbers a; b; and c; If a = b; then a + c = b + c
:
Example 2: (Using the addition property of equality)
Solve and check your answer.
3
2
a)
=x
5
5
b)
Page: 1
6:2 + x =
1:89
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
2.1
The Subtraction Property of Equality:
For any real numbers a; b; and c; If a = b; then a
c=b
c
:
Example 3: (Using the addition property of equality)
Solve and check your answer.
20
a) b + 7 =
3
b)
1
5
= +x
4
6
The Multiplication Property of Equality:
For any real numbers a; b; and c; c 6= 0, If a = b; then ca = cb
:
Example 4: (Using the multiplication property of equality)
Solve and check your answer.
x
a)
=5
b)
2
s
=
3
5
6
The Division Property of Equality:
For any real numbers a; b; and c; c 6= 0, If a = b; then
a
b
=
c
c
:
Example 5: (Using the division property of equality)
Solve and check your answer.
5
a) 5x =
8
b)
Page: 2
32w = 64
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
2.1
Markdown and Markup
When the price of merchandise is reduced, the amount of reduction is called the markdown or the discount. To …nd
the sale price of an item, we subtract the markdown from the regular price.
Example 6: (Solving markdown problems)
A sofa is on sale for $650. If it has been marked down $325, …nd its regular price.
Example 7: (Solving markdown problems)
A house that was priced at $105,000 has been discounted $7,500. Find the new asking price.
To make a pro…t, a merchant must sell an item for more than he paid for it. The retail price of the item is the sum
of its wholesale cost and the markup.
Example 8: (Solving markup problems)
A car with a sticker price of $17,500 has a mark up of $3,500. Find the invoice price (the wholesale price) to the dealer.
Page: 3
Notes by Bibiana Lopez
Beginning and Intermediate Algebra by Gustafson and Frisk
2.1
Example 9: (Solving markup problems)
A vacuum that sells for $97 has a markup of $37. Find the wholesale price.
Percent
Percent Formula:
If r is the rate, b is the base, and a is the amount, then rb = a
:
Example 9: (Using the percent formula)
Use the formula rb = a to …nd each value.
a) What percent of 540 is 81?
b) 16% of what number is 40?
Page: 4
c) 10% of 80 is what number?
Notes by Bibiana Lopez