Transcript Ch 3 Alg 1 07-08 KS, EN

Chapter 3
Solving Linear Equations
By
Kristen Sanchez and Emma Nuss
INVERSE OPERATIONS:
• Two operations that undo each other, such
as addition and subtraction are called
inverse operations. These can help you
isolate the variable on one side of the
equation.
• To solve an equation with a fractional
coefficient, such as 10 = 2/3m, multiply
each side of the equation by the reciprocal
of the fraction (in this case 3/2). The
product of a nonzero number and its
reciprocal is 1.
Summary: Properties of Equality
• Addition Property of Equality:
If a = b, then a + c = b + c
• Subtraction Property of Equality:
If a = b, then a – c = b – c
• Multiplication Property of Equality:
If a = b, then ca = cb
• Division Property of Equality:
If a = b and c does not = 0, then a/c = b/c
3.1: Solving Equations Using
Addition and Subtraction
Operation
Original
Equation
Add the same
x-3=5
number to each
side
Subtract the
x+6=10
same number
from each side
Simplify one or x=8-3
both sides
Equivalent
Equation
x=8
x=4
x=5
3.2: Solving Equations Using
Multiplication and Division
Operation
Original
Equation
Equivalent
Equation
Multiply each
side of by the
same nonzero
number.
x/2=3
X=6
Divide each
side of by the
same nonzero
number.
4x=12
X=3
Solving Multi-Step Equations
• When solving a multi-step equation, the
first step is isolate the variable, then solve
for x.
Example:
3x+7=-8
-7 -7
3x=-15
x=-5
Solving Equations with Variables on
Both Sides
• Some equations have variables on both sides. To solve
these, you collect the variable terms on one side of the
equation.
• Example:
7x+19=-2x+55
+2x +2x
9x+19=55
-19 -19
9x=36
9 9
x=4
Tips on Solving Linear Equations
• Simplify: combine like terms or distribution
• Collect: put the variable terms on the side
with the larger coefficient
• Inverse operations: use inverse operations
to isolate the variable
• Check: check your solution with the
original equation
3.6: Solving Decimal Equations
• If you have a long decimal answer to an equation, it is
often more practical to round to an approximate answer
• Example:
-38x-39=118
+39 +39
-38x=157
x=157/-38
x≈-4.131578947
x≈-4.13
• The rounded answer may not be exactly equal to the
original equation, but the symbol ≈ is used to show that
the answer approximately balances the two sides of the
equation.
3.7: Formulas
• Formulas are algebraic equations that relate two or more
quantities.
• Formulas can be used to solve problems with variables
other than x and y.
• Example:
Celsius and Fahrenheit are related by the equation
C=5/9(F-32) where C is Celsius and F is Fahrenheit.
Solve the formula for degrees Fahrenheit F.
C=5/9(F-32)
*9/5 *9/5
9/5C=F-32
+32 +32
9/5C+32=F
3.8: Ratios and Rates
• The ratio of a to b is a/b.
• If a and b are measured in different units,
then a/b is called the rate of a per b.
• Rates are often expressed as unit rates, or
the rate per one given unit (ex. 60 miles
per 1 gallon)
Unit Analysis
• Writing the units when comparing each
quantity of a rate is called unit analysis.
• You can multiply and divide units just as
you can multiply and divide numbers.
• Example:
60 mins= 1 hour
convert 3 hours to minutes
3 hours x 60 mins/1 hour= 180 mins
3.9: Percents
• A percent is a ratio that compares a number to 100. For
example, forty percent can be written as 40/100, 0.40, or
40%.
• Number being compared to base= a
percent= p/100
base number = b
a = p/100 * b
• Example:
Fourteen dollars is 25% of what amount of money?
Percent = 25/100 =1/4
14 = ¼ * x
4(14) = 4(1/4) * x
56 = x
Using Percents to find Discounts
• Discount is the difference between the regular price of
and item and its sale price. To find the discount percent,
use the regular price as the base number in the percent
equation.
• Example
A portable CD player has a regular price of $90 and
a sale price of $72. What is the discount percent?
Discount=regular price-sale price
=90-72=18
Percent=p/100
Regular price=90
18=p/100 * (90)
18/90=p/100
.20=p/100
20=p
Summary of Percents
a = p/100 * b
Question
Given
Need to Find
What is p
percent of b?
b and p
Number
compared to
base, a
Base number, b
a is p percent of a and p
what?
a is what
percent of b?
a and b
Percent, p