Transcript Chapter 7 Slides

Chapter 7: Basic Concepts of
Algebra
7.1
Linear Equations
• An equation in the variable x is linear if it
can be written in the form
Ax + B = C
where A,B,C are real numbers and A is not
0.
7.1
Helpful properties
• Addition property:
A = B and A + C = B + C
are equivalent (same solutions).
• Multiplication property:
A = B and AC = BC
Are equivalent as long as C is not 0.
7.1
Solving a linear equation
1.
2.
3.
4.
Clear fractions
Simplify each side separately
Isolate the variable terms on one side
Transform so that the coefficient of the
variable is 1
5. Check your solution
7.1
Kinds of Linear equations
• Conditional: finite number of solutions
Ex. 2x = 4
• Contradiction: no solutions
Ex. 2x + 1 = 2x +5
• Identity: true for any number
Ex. 2x + 2 = 2(x + 1)
7.2
Applications of Linear Equations
Key Words
Operation
Sum, more than, plus,
added, increased
Addition
Less than, difference,
minus, decreased
Subtraction
Times, multiplied by,
product
Multiplication
Quotient, ratio, divided
by
Division
7.2
Examples
If a quotient of a number and
6 is added to twice the
number, the result is 8 less
than the number
7.2
Solving an Applied Problem
1. Read the problem carefully
2. Assign a variable to the unknown value,
and write down any other unknowns in
terms of this variable. Use tables,
diagrams, etc.
3. Write equation using the variable.
4. Solve the equation.
5. State the answer in words relative to
the context. Is it reasonable?
6. Check the answer in the words of the
original problem.
7.2
Coin Mixture
Dave collects US gold coins. He
has a collection of 80 coins. Some
are $10 coins and some are $20
coins. If the face value of the coins
is $1060, how many of each
denomination does he have?
7.2
Alcohol Mixture
How many liters of a 20%
alcohol solution must be
mixed with 40 liters of a 50%
solution to get a 40%
solution?
Distance, Rate and Time
7.2
#62 Time Traveled on a Visit
Steve leaves Nashville to visit his
cousin Dave in Napa, 80 miles away.
He travels at an average speed of 50
miles per hour. One half-hour later
Dave leaves to visit Steve, traveling at
an average speed of 60 miles per hour.
How long after Dave leaves will they
meet?
#57 Travel Times of Trains
A train leaves Little Rock,
Arkansas, and travels north at 85
km/hr. Another train leaves at the
same time and travels south at 95
km/hr. How long will it take before
they are 315 km apart?
7.3
Ratio, Proportion and Variation
7.3
Proportion and Cross Products
#26: Proper Dosages
A nurse is asked to administer 200
milligrams of fluconazole to a
patient. There is a stock solution
that provides 40 milligrams per
milliliter. How much of the solution
should be given to the patient?
Golden Ratio/Divine Proportion
Fibonacci sequence: 1, 1, 2, 3, 5, 8, …
What happens to ratios of successive
Fibonacci numbers?
Golden ratio is 1.618…
Fibonacci Squares
7.3
Direct Variation
#44 Paddleboat Voyage
According to Guinness World Records, the
longest recorded voyage in a paddleboat is
2226 miles in 103 days by the foot power of
two boaters down the Mississippi River.
Assuming a constant rate, how far would
they have gone if they had traveled 120
days? (Distance varies directly as time)
7.3
Direct Variation as a Power
#51 Falling Body
For a body falling freely from rest,
the distance the body falls varies
directly as the square of the time.
If an object is dropped from the top
of a tower 400 feet high and hits
the ground in 5 seconds, how far
did it fall in the first 3 seconds?
7.3
Inverse Variation
Joint and Combined Variation
If one variable varies as the product of
several other variables, it is said to be in
joint variation.
Combined variation involves combinations
of direct and inverse variation.
#53 Skidding Car
The force needed to keep a car from
skidding on a curve varies inversely as the
radius of the curve and jointly as the weight
of the car and the square of the speed. If
242 pounds of force keep a 2000 pound car
from skidding on a curve of radius 500 feet
at 30 miles per hour, what force would keep
the same car from skidding on a curve of
radius 750 feet at 50 miles per hour?
Properties of Exponents
7.5
Power Rules for Exponents
7.6
Special Products of Polynomials
7.7
Quadratic Equations
•An equation of the form
ax2 + bx + c = 0
where a,b,c are real numbers with a not
equal to 0, is a quadratic equation.
• Zero factor property (helpful for
factoring):
If AB = 0, then A = 0 or B = 0 or both
Quadratic Formula
7.7
Applications
The Toronto Dominion Centre in Winnipeg,
Manitoba is 407 feet high. Suppose that a
ball is projected upward from the top of the
centre and its position s in feet above the
ground is given by the equation s = -16t2 +
75t + 407, where t is the number of seconds
elapsed. How long will it take for the ball to
reach a height of 450 feet?
7.7 #68, page 355
• A club swimming pool is 30 feet wide by
40 feet long. The club members want a
border in a strip of uniform width around
the pool. They have enough material for
296 square feet. How wide can the strip
be?