9/23/11 4.1 & 4.2 Systems of Equations
•  Determine if ordered pairs are solutions of systems of
equations.
•  Solve 2x2 linear systems
•  Graphically
•  Using the method of substitution
•  Using the method of elimination
•  Use systems of equations to model and solve application
problems.
Bork got 18 on the first quiz and improves by 2 points
each week. Brenda got 15 on the first quiz and
improves by 3 each week. On which quiz will they
have equal scores?
1 9/23/11 Quick draw
What does it mean to a solution to system
of two linear equations?
CASE 1
CASE 2
CASE 3
2 9/23/11 •  System of equations is a set of two or more equations
with two or more unknowns.
•  Example:
•  A solution to the system is an ordered pair (x, y) that
satisfies both equations.
A fundraising dinner was held on two consecutive nights. On the
first night, 100 adult tickets and 175 children’s tickets were sold,
for a total of $950. On the second night, 200 adult tickets and 315
children’s tickets were sold, for a total of $1830. Find the price of
each type of ticket.
3 9/23/11 Ten pounds of mixed nuts sells for $9 a pound. The mixture
is obtained from two kinds of nuts, peanuts priced at $3 a
pound and cashews at $13 per pound. How many pounds
of each variety of nuts is used in the mixture?
You need to do your laundry. The Laundry Fairy will
give you all of her quarters, if you can solve her riddle:
“I have $5.00 in nickels and quarters. The total number
of coins is 56. How many quarters do I have for you?”
4 9/23/11 Moving Soon?
Which is the better value when renting a vehicle?
U-Haul charges $29.95 plus $0.89 per mile.
Penske charges $34.95 per day plus $0.32 per mile.
Examples
⎧ −x + y = 5
⎨
⎩2(x −1) − 2y = 5
€
5 9/23/11 Examples
⎧2x − 4 y = 9
⎨
⎩ x − 4.5 = 2y
€
Examples
⎧ 1
1
⎪ x + y = 8
⎨ 5
2
⎪⎩ x + y = 20
€
6 9/23/11 Example
⎧2x − 3y = 4
⎨
⎩ 4 x + 2y = 2
€
Example
7 9/23/11 Differences
•  SUBSTITUTION METHOD:
•  Goal is to express one variable in terms of the other and
substitute the new value in the second equation to
obtain one equation in one variable
•  ELIMINATION METHOD:
•  Goal is to have one variable with opposite coefficients,
so that adding the two equations will eliminate that
unknown from the equation and lead to one equation in
one variable
4.3 Linear Systems in Three Variables
•  Solve 3x3 systems of linear equations.
8 9/23/11 Remember the Laundry Fairy?
Well now she has dimes, too.
•  “I have $5.00 in nickels, dimes, and quarters. There are
50 total coins, and twice as many dimes as nickels. How
many of each coin do I have?”
Example
⎧2x − 2y + z = 4
⎪
⎨
y + 2z = 4
⎪
z =2
⎩
€
9 9/23/11 Example
⎧ x + y + z = 6
⎪
⎨2x − y + z = 3
⎪
-z =0
⎩ 3x
€
Example
10 9/23/11 A chemist needs 100 liters of a 25% acid solution. It is mixed from
three solutions whose concentrations are 10%, 20%, and 50%.
And the same amount of 10% and 50% solution must be used.
How many liters of each solutions should be used to mix the
solution?
11