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December 9th

December 9th

# Systems of Linear Equations

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

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

2x + y = 2
4x + 2y = –5

2x + 4y = 7
5x – 3y = –2

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.

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