Chapter 6
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Two equations that represent two different
relationships with respect to the same two
unknown variables
. Ex: set up two different equations to model
the two different relations:
 The cost to begin production on a band’s CD is
$1500. Each CD costs $4 to produce and will sell
for $10 each. The band wants to know how many
CDs they will have to sell to earn a profit
INCOME


Let x = number of cds
Y = 10x
EXPENSES


Let x = number of cds
Y = 1500 + 4x

Y = 10x

Y = 1500 + 4x


when the x and y value(s) satisfy both
equations, ie; When you “plug in” the values
for the variables the equation balances
On a graph, when the lines representing the
relations share a common point(s)

One Solution
 lines intersect
 Slopes are the different
 Relationship is consistent and independent

Infinite Solutions
 Lines are co-linear~ one equation has been scaled!
 Slopes will be the same (when equation is scaled)
 Relationship is consistent and dependent

No Solution
 Lines are parallel
 Slopes are the same
 Relationship is inconsistent
1.) y = -2x +3
y=x–5
2.) y = 2x + 3
y = -2x – 5
3.) x – 2y = 4
x = 2y - 2
4.) x + 2y = 1
2x + 5 = y
5.) 3x – y = 2
12x – 4y = 8
Real Life
example…
High School Sport
Number of girls
participating in 2008 (in
thousands)
Average rate of increase
(thousands per year)
soccer
345
8
Track and field
458
3
Write and graph a system of equations to predict when the participation in the two sports
will be the same
Let x = number of years since 2008 and y = number of girls competing
Soccer:
Track and Field:
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

Joe and Josh each want to buy a video game.
Joe has $14 and saves $10 per week. Josh has
$26 and saves $7 per week. In how many
weeks will they have the same amount?
Let x =
Let y =
1.) y = 3x – 4
y = -3x + 4
3.) y = -6
4x + y = 2
2.) x + 2y = 5
2x + 4y = 2
4.) 2x + 3y = 4
-4x – 6y = -8
* Check with desmos
Graphing (not the most efficient)
Substitution
 Elimination
 Addition and Subtraction
 Multiplication and Division


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Solve one equation for the variable with a
coefficient of 1 or -1 (or for the coefficient
that is easiest to solve for)
Substitute the expression equal to the
variable into the other equation and solve for
the remaining variable
Substitute this variable into either equation
to find the value for the original variable.
The resulting point is the solution
1.) x + 4y = 1
2x – 3y = -9
3.) 5/2x + y = 4
5x + 2y = 8
2.) 3x + 4y = 7
1/2x + y = 11
4.) 2x – y = 8
y = 2x - 3

A store sold a total of 125 car stereo systems
and speakers in one week. The stereo
systems sold for $ 104.95, and the speakers
sold for $18.95. The sales from these two
items totaled $6926.75. How many of each
item were sold?
EJH Labs needs to make
1000 gallons of a 34%
acid solution. The only
solutions available are
25% acid and 50% acid.
How many gallons of
each solution should
be mixed to make the
34% solution?
A metal alloy is 25%
copper. Another metal
alloy is 50% copper.
How much of each
alloy should be used to
make 1000 grams of
metal alloy that is 45%
copper?

Use this method when
one of your variables
have the same
coefficients or the
coefficients are
additive inverses
(opposites)
Write the equations so
the variables are
aligned
 Add or subtract the
equations to eliminate
one variable
 Substitute and solve
for the other variable
 Write your solution as
an ordered pair

1.) 4x + 6y = 32
3x – 6y = 3
2.) -4x + 3y = -3
4x – 5y = 5
3.) 4y + 3x = 22
3x – 4y = 14
The sum of two numbers
is 18. The sum of the
greater number and
twice the smaller
number is 25. Find the
numbers.

The sum of two
numbers Is 27. Their
difference is 5. Find
the numbers.

Lena is preparing to take the SATs. She has
been taking practice tests for a year and her
scores are steadily improving. She always
scores about 150 higher on math than she
does on verbal. She needs a 1270 to get into
the college she has chosen. If she assumes
that she will still have that 150 difference
between the two tests, what will she have to
score on each part?
 Use this method
when both
variables have
different
coefficients




Scale one of the
equations to make the
coefficient of one
variable either the same
or the opposite of that
variable of the other
equation
Add or subtract the
equations to eliminate
the one variable.
Substitute and solve for
the other variable
Present your answer as
an ordered pair
1.) 2x + 3y = 5
5x + 4y = 16
2.) 3x + 5y = 11
2x + 3y = 7
3.) 2x – 3y = 8
-5x + 2y = 13
4.)
6a + 2b = 2
4a + 3b = 8
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
A personal aircraft traveling with the wind
flies 520 miles in 4 hours. On the return trip,
the airplane takes 5 hours to travel the same
distance. Find the speed of the plane.
Let a= rate of airplane in still air
Let w= rate of the wind

A bank teller reversed the digits in the
amount of a check and overpaid the
customer by $9. The sum of the digits in the
two digit amount was 9 Find the amount of
the check.

Graphing
 To estimate solutions

Substitution
 One one of the variables in either solution has a
coefficient of 1 or -1

Elimination with Addition


Elimination with Subtraction


If one of the variables has opposite coefficients in the two
equations
If one of the variables has the same coefficient in the two
equations
Elimination with Multiplication
 Anytime!
1.) 4x – 4y = 8
-8x + y = 19
2.) 5x +7y = 2
-2x + 7y = 9
3. ) x – y = 9
7x + y = 7
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
Of the 17 species of penguins in the world,
the largest species is the empire. One of the
smallest is the Galapagos penguin. The total
height of the two penguins is 169
centimeters. The emperor penguin is 22 cm
more than twice the height of the Galapagos
penguin. Find the height of each.
Let e =
Let g =


These systems can have a range of solutions
represented by an overlap of shading one a
graph
OR not solution represented by a nonintersecting region on a graph

1.) y < 3
x+y>1

3. ) 3x – y > 2
3x – y < -5

2.) 2x + y > 2
2x + y < 4

4.) y < 2x + 2
y > -x - 3

The Theatre Club is selling shirts. The have
only enough supplies to print 120 shirts. They
will sell sweatshirts for $22 and T shirts for
$15, with a goal of at least $2000 in sales.
 Define variables
 Graph the system
 Name one possible solution
 Is (45,30) a solution?