6.1 Solving Systems of Linear
Equations by Graphing
Equations considered together are
called a system of equations.
A solution of a system of equations in
two variables is an ordered pair that
is a solution of each equation of the
system.
2x + y = 3
x + y = 1
(2, -1)?
(3, -3)?
Is (1,-3) a solution of the system
3x + 2y = -3
x – 3y = 6?
Is (-1,-2) a solution of the system
2x – 5y = 8
–x + 3y = –5?
The solution of a system of linear
equations in two variables can be
found by graphing the two equations
on the same coordinate system .
Independent
Inconsistent
Dependent
Solve by graphing:
2x + 3y = 6
2x + y = –2
Solve by graphing:
2x – y = 1
6x – 3y = 12
Solve by graphing:
x – 2y = 2
x + y = 5
Solve by graphing:
4x – 2y = 6
y = 2x – 3
Solve by graphing:
x + 3y = 3
–x + y = 5
Solve by graphing:
6x – 2y = –6
y = 3x – 1
6.2 Solving Systems of Linear
Equations by Substitution Method
Solve by substitution:
2x + 5y = –11
y = 3x – 9
Solve by substitution:
5x + y = 4
y = –5x + 4
Solve by substitution:
y = 3x – 1
y = –2x – 6
Solve by substitution:
3x – y = 4
y = 3x + 2
Solve by substitution:
7x – y = 4
3x + 2y = 9
Solve by substitution:
y = –2x + 1
6x + 3y = 3
For what value of k does the system
have no solution
y = –2x + 1
2y = kx + 2
Rewrite each equation so the
coefficients and constant are integers
.7x – .1y = .4
.3x + .2y = .9
6.3 Solving Systems of Linear
Equations by Addition Method
3x + 2y = 4
4x – 2y = 10
Sometimes you need to multiply first
3x + 2y = 7
5x – 4y = 19
Sometimes you need to multiply both eq.
5x + 6y = 3
2x – 5y = 16
5x = 2y – 7
3x + 4y = 1
Solve by the addition method:
2x + y = 2
4x + 2y = –5
2x + 4y = 7
5x – 3y = –2
Solve by the addition method:
x – 2y = 1
2x + 4y = 0
2x – 3y = 4
–4x + 6y = –8
6.4 Application Problems in Two
Variables
Rate-of-wind and water–current
problems
• Choose one variable to represent
the rate of the object in calm
conditions
• Choose second variable to
re present rate of wind or current
• Use these variables to express
the rate of object with (or without)
the wind or current
• Use these variables to express
the rate of object against (or with)
wind or current
• Create table: Rate x time = distance
• Set each variable expression for
distance equal to known distance.
• Solve for one variable
• Substitute and solve for second variable
Flying with the wind, a small plane
can fly 750 mi in 3 h. Against the
wind, the plane can fly the same
distance in 5 h. Find the rate of the
plane in calm air and the rate of the
wind.
A 600-mile trip from one city to
another takes 4 h when a plane is
flying with the wind. The return trip
against the wind takes 5h. Find the
rate of the plane in still air and the
rate of the wind.
A canoeist paddling with the current
can travel 24 mi in 3 hrs. Against the
current, it takes 4 h to travel the
same distance. Find the rate of the
current and the rate of the canoeist in
calm water.
Application Problems
• Choose one variable to represent
one unknown quantity
• Choose second variable to
represent other unknown quantity
• Write numerical or variable
expressions for all remaining
quantities
• Record results in two tables
• De termine an equation from the first
table
• Determine a second equation from
the second table
• Solve for one variable
• Substitute and solve for second
variable
A jeweler purchased 5 oz of a gold
alloy and 20 oz of a silver alloy for a
total cost of $700. The next day, at
the same prices per ounce, the
jeweler purchased 4 oz of the gold
alloy and 30 oz of the silver alloy for
a total cost of $630. Find the cost
per ounce of the silver alloy.
A storeowner purchased 20
incandescent light bulbs and 30
fluorescent bulbs for a total cost of
$40. A second purchase, at the
same prices, included 30
incandescent bulbs and 10
fluorescent bulbs for a total cost of
$25. Find the cost of an
incandescent bulb and of a
fluorescent bulb.
Two coin banks contain only dimes
and quarters. In the first bank, the
total value of the coins is $3.90. In
the second bank, there are twice as
many dimes as in the first bank and
one-half the number of quarters . The
total value if the coins in the second
bank is $3.30. Find the number of
dimes and the number of quarters in
the first bank.
