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.