**6.1 Solve Systems of Linear Equations by Graphing
6.2 Solve Systems of Linear Equations by the Substitution
Method
6.3 Solve Systems of Linear Equations by the Addition
(Elimination) Method
6.4 Application Problem Involving Two Equations and Two
Variables.**

A **system of equations** is a set of two or more
equations

considered together. A s**olution to a system of equations** in two

variables is an ordered pair that is a solution to every equation in

the system.

**Example 1 Solutions to a System of Equations**

a. Is (2, -1) a solution to the system

b. Is (1, -3) a solution to the system

Recall: All ordered pair solutions to an equation form**
the graph
of an equation in two variables.**

**Example 2 Solve by the Graphing Method**

a. Solve by graphing

b. What is the point of intersection of

the two graphs?

c. What is the solution to the system of

equations?

d. What is the relationship between the point(s) of
intersection and

the solution(s) to the system of equations.

**The Solution Set to a Linear System of Two Equations
in Two Variables has Three Possible Outcomes . . .**

**1. One Solution **See Figure 1. The lines

intersect at one point and the solution set

contains only one point. A system of linear

equations with at least one solution is

called a consistent system. The solution to

the system is expressed as an ordered pair

(a,b) .

2. **No Solutions** See Figure 2. The lines

are distinct and parallel, and never

intersect. Therefore, there is no solution.

A linear system whose solution set is

empty is called an inconsistent system.

3.** Infinite Number of Solutions** See

Figure 3. The lines lie on top of each other,

are not distinct, and have an infinite number

of points of intersection. Therefore, the

system of equations has infinite solutions.

This is called a dependent system.

**Example 3 An Inconsistent System means the system
has no solution and the two lines are parallel.**

Solve by graphing

**Example 4 A Dependent System means the two lines are
the same.**

Solve by graphing

**Example 5 **

Solve by graphing

**Example 6**

Solve by graphing

**Example 7 **

Solve by graphing

**Example 8 **

Solve by graphing

Solve

**Algebraic Methods to Solve Systems of Equations
**The solution to the above system is (-3/4, -3/4). The graphing

method only works well when the ordered pair solution contains

only integer numbers. Therefore, we need an algebraic method for

solving systems of equations. The two algebraic methods we are

going to learn are the

(Elimination) Method.

**Substitution Method**

1. Isolate one variable in one of the given equations.

2. Substitute the equation from step 1 into the other equation and

solve for the one variable remaining in that equation.

3. Use the equation from step 1 to find the value of the other

variable.

4. Write your solution as an ordered pair.

5. Check the solution.

**Example 1**

Solve by substitution

**Example 2**

Solve by substitution

**Example 3**

Solve by substitution

**Example 4 **

**Solve** by substitution

**Example 5 **

Solve by substitution

**Two exceptions that may occur when using the
Substitution Method**

**Inconsistent System** If at some point in working the
algebra in

step 2 you get an untrue statement, such as 3 = 5, then you have an

inconsistent system. That is, the graphs of the lines are parallel

and never intersect.

**Dependent System** If at some point in working the
algebra in

step 2 you get a worthless but true statement, such as 4 = 4, then

you have a dependent system. That is, the graphs of the lines lie

on top of each other and the system has an infinite number of

solutions (points of intersection).

**The Solution Set to a Linear System of Two Equations
in Two Variables has Three Possible Outcomes . . .**

1.** One Solution **See Figure 1. The lines

intersect at one point and the solution set

contains only one point. A system of linear

equations with at least one solution is

called a consistent system. The solution to

a consistent system is expressed as an

ordered pair (a,b) .

2. **No Solutions **See Figure 2. The lines

are distinct and parallel, and never

intersect. Therefore, there is no solution.

A linear system whose solution set is

empty is called an inconsistent system. The

Elimination and Substitution methods

produce a false statement, such as 4 = 0.

3. **Infinite Number of Solutions** See

Figure 3. The lines lie on top of each other,

are not distinct, and have an infinite number

of points of intersection – every point on

the line. Therefore, the system of equations

has an infinite number of solutions. This is

called a dependent system. The Elimination

and Substitution methods produce a true

statement, such as 3 = 3.

**Example 6**

Solve by substitution

**Example 7**

Solve by the substitution method

**Example 1**

Solve by the Addition Method

Recall **If** c ≠ 0 and a = b , then ac = bc .

This means you can multiply both sides of an equation by

the same non-zero number without changing the solution

set (i.e. the graph) of the equation. The modified system

of equations is equivalent to the original system.

**Example 2 **

Solve by the Addition Method

**Example 3 Both equations may have to be changed**

Solve by the Addition Method

**Note To use the addition method both equations must be
written so that all variable terms are on one side of
each equation and the constant term is on the other
side.**

**Example 4 **

Solve by the Addition Method

**Example 5 **

Solve by the Addition Method

**Example 6**

Solve by the Addition Method

**Example 7**

Solve by the Addition Method

**Example 8 **

Solve by the Addition Method

**Example 1**

Flying with the wind, a plane can fly 750 miles in 3 hours. Against

the wind, the plane can fly the same distance in 5 hours. Find the

rate of the plane in calm air and the rate of the wind.

**General Strategy**

1. Choose one variable to represent the rate of the object in a calm

condition and a second variable to represent the rate of the

wind or current.

let

p = the rate of the plane in calm air

w = rate of the wind

so that

p + w = rate of the plane traveling with the wind

p - w = rate of the plane traveling against the wind

2. Use the expressions from step 1, as well as the time
traveled

with and against the wind, to get expressions for the distances

traveled with and against the wind. Recall:

Rate ยทTime = Distance is used to determine the expression
for the

distance traveled. Organize the data in a table.

Trip |
Rate |
. |
Time |
= |
Distance |

With the wind | . | = | |||

Against the wind | . | = |

3. Write a system of equations that models the problem.

4. Solve the system of equations and answer the question.

**Example 2**

A 600 mile trip from one city to another takes 4 hours when the

plane is flying with the wind. The return trip against the wind

takes 5 hours. Find the rate of the plane in still air and the rate of

the wind.

**Example 3
**A canoeist paddling with the current can travel 24 miles in 3 hours.

Against the current, it takes 4 hours to travel the same distance.

Find the rate of the canoeist in calm water and the rate of the

current.

Equations and Two Unknowns

**Example 4**

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.

**Example 5**

A store owner purchases 20 incandescent light bulbs and 30

fluorescent bulbs for $40. A second purchase, at the same prices,

included 30 incandescent light bulbs and 10 fluorescent bulbs for

$25. Find the cost of each type of bulb.

**Example 6**

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 of the coins in the second bank is

$3.30. Find the number of dimes and quarters in the first bank.