Systems of Equations
4.1 Graphing
4.2 Substitution
4.3-4.4 Elimination
4
To solve a system by graphing:
Solve for y,
Make a table,
use slope &
plot points
y-intercept
Find the x- &
y-intercepts
1. Graph both lines.
2. Write the point where they cross as an ordered pair.
3. Check the solution by plugging in x & y into both equations.
1 solution
(2, 1)
no solution
infinitely many solutions
Ways to graph:
SECTION 4.1: SOLVING SYSTEMS
USING GRAPHING
Solve by graphing.
Dy
ne
b) 2y = -x + 4 and y + 3x = -3
z
E
a) y = 2x - 3 and y = x - 1
m
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cn
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3
3
Without graphing, determine how many solution
each system has.
a) y = 3x + 2 ; y - 3x = -2
5
Insolation
b) 3x + 4y = 12 ; y = -3/4x + 3
3x
3X
15
7
infinitutionO
c) y = 2x + 7 ; y = x + 6
onesolution
3
X
4
5
vertical
horizontal
line
line
SECTION 4.2: SOLVING SYSTEMS
USING SUBSTITUTION
Define.
a) system of linear equations: Two or more linear equations
together.
b) solution of the system of linear equation: is an ordered
pair (x,y) that satisfies both equations.
Number of solutions.
1 solution
o
(2, 1)
no solution
infinitely many solutions
Substitution Method:
The process of replacing one variable with an equivalent
expression containing the other variable.
Steps:
a. make sure 1 equation is solved for x or y
b. substitute for x (or y) & solve for the
variable that's left
c. plug that answer into original equation to
find the other variable
d. Write your solution as an ordered pair in
alphabetical order.
Check your solution in both equations.
Just Watch!
2x - y = 3 and x - y = 1
Tf
zCity y 3 x
2 2 y 17 3
ztly
Z
3
z
tyty
l
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x
2
cz
Steps:
a. make sure 1 equation is
solved for x or y
b. send in a substitute
for x (or y) & solve for
the variable that's left
c. plug that answer into
an original equation to
find the other variable
d. Write your solution as
an ordered pair in alpha.
order. Check your
solution in both equations.
Solve for both variables using the substitution method.
a) a - b = 4
a 1 2
1a
Ga
b)
54
6
b
4
4
2t5a
2
b
and b = 2 - 5a
4
ia
x - 2y = 6 and 3x + 2y = 4
EEE
ah
c) y - x = 5 and 2x + y = 8
Tix
y
f5tx
4 5 1
y 6
1
15 4 8
2
2
3
514
5
5
8
8
5
3,1 3
I
z
2
2
Ta
54
5
3
Solve for both variables using the substitution method.
d) 2x + y = 13
4x - 3y = 11
To
e) y = 2x + 2 y = -3x + 4
2
3 1 4
2
3k
3x
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4
if
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SECTION 4.3-4.4: THE ELIMINATION METHOD
Elimination Method: The process of combining two
equations in order to "eliminate" one of the variables.
Does the system contain opposite coefficients?
YES
1) Combine like terms
down the columns.
2) Solve for remaining
variable.
3) Substitute that value
into either original
equation.
4) Solve for the other
variable.
5) Write answers as an
ordered pair.
NO
1) Determine which
variable is easiest to
eliminate.
2) Determine what the
coefficients need to be to
eliminate.
3) Multiply one or both
equations to create
opposite coefficients.
4) Follow steps 1 - 5 from
the YES column
Solve each system of equations using the elimination
method.
T
2
2 1
21
z
t
44 68
I
Ty
h
y
8
T
8X
10
54 17
54 17
17
54
If
4
Z
3
c)I x + ly = 5
2x - y = 4
b) 2x - 3y = 61
-2x - y = 7
a) 2x + 5y = 17
t 6x - 5y = -9
2x
2
2x
X
c
J
4 7
f 171 7
2
17
17
7
17
10
I
5
7
3
3
5
5
Xt 1
4
3
3
4
9
J
3
2
Solve each system of equations using the elimination
method.
HARDER....
I'll
21
d) 6x + y = 18
3x + 2y = 18
t
f2
12
lyC
2
a
21 181
36
18
24
18
9
6
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18
4
18
4
18
4
12
12
y
I
2
X
12
6
2
93
1
9
e) x + y = -7It
3x + y = -9
Ixtittlyc D 7C t
Ix
ly _7
3
14
ex
2
xty
it 1
y
2
IT
7
7
I
6
Solve each system of equations using the elimination
method.
HARDEST....
f) 3x + 5y = 30
5x + 8y = 49
g) 6x + 2y = 4
10x + 7y = -8
h) 2x + 5y = -22
10x + 3y = 22
1. Suppose you have $20 in your bank account and you start
saving $5 each week. Write an expression that represents
this situation.
2. Your friend has $5 in his account and is saving $10 each
week. Write an expression that represents this situation.
3. Determine how many weeks it will take for you and your
friend to have the same amount of money.
SECTION 4.5: USING A SYSTEM OF EQUATIONS
Set up & solve a system of equations for each problem.
1) The sum of two numbers is 115 and a difference of 21. Find
the numbers.
2) The sum of two numbers is 27. One number is 3 more than
the other. Find the numbers.
Set up & solve a system of equations for each problem.
3) Ramon sells car and trucks. He has room on his lot for 510
vehicles. From experience he knows that his profits will be
greatest if he has 190 more cars than trucks. How many of each
vehicle should he have?
4) A class has 31 students. There are 5 more boys than girls.
How many girls are there?
Set up & solve a system of equations for each problem.
5) Bob is 6 years older than Fred. Fred is half as old as Bob. How
old are they?
6) Suzy and Molly scored a total of 21 points. If Suzy scored 6
more than twice what Molly scored, how many points did they each
score?
USING A SYSTEM OF EQUATIONS
Set up & solve a system of equations for each problem.
1) Apples cost $0.40 each and pears cost $0.55 each. Anthony
bought a total of 13 pieces of fruit for a total cost of $5.80. How
many apples and how many pears did Anthony buy?
2) There were 429 people at a play. Admission was $1 for adults
and $0.75 for children. The total earned was $372.50. How many
adults and how many children attended the play?
Set up & solve a system of equations for the problem.
3) Your school is planning on bringing 193 people to a competition at
another school. There are 8 drivers available and two types of vehicles,
school buses and minivans. The school buses seat 51 people each and
the minivans seat 8 people each. How many buses and minivans will be
needed?
Set up & solve a system of equations for each problem.
4) Kara has 13 coins in her pocket, some nickels and some dimes. The
value of the coins is $1.00. How many nickels and how many dimes does
she have?
5) There are 20 coins on the table, some dimes and some quarters.
Their value is $3.05. How many of each type are there?
Set up & solve a system of equations for each problem.
6) You have $22 in your bank account and deposit $11.50 each
week. At the same time, your brother has $218 but is withdrawing
$13 each week.
a) When will your accounts have the same balance?
b) How much will each of you have after 12 weeks?
Set up & solve a system of equations for the problem.
7) The perimeter of a rectangle is 34 cm. The perimeter of the
triangle is 30 cm.
x
y
x
y+1
y