# 7 5 Solve Special Types of Linear Systems A by 35HFP4XV

VIEWS: 24 PAGES: 23

• pg 1
```									7.5 Solve Special Types of
Linear Systems

You will identify systems that have no solutions
and systems that have infinitely many
solutions.

Essential Question: How can you identify the
number of solutions in a linear system?
Warm-Up Exercises      Lesson 7.5, For use with pages 459-465

1. Solve the linear system.
2x + 3y = – 9
x – 2y = 6

2. You buy 8 pencils for \$8 at the bookstore. Standard
pencils cost \$.85 and specialty pencils cost \$1.25.
How many specialty pencils did you buy?

Independent Practice Check
Textbook p. 447-450
Independent Practice Check
Textbook p. 454-457
Vocabulary
       Exactly One Solution = Consistent and Independent
     Graphing – Intersecting Lines
     Substitution – Produces a Solution
     Elimination – Produces a Solution

       Infinite Solutions = Consistent and Dependent
     Graphing – Same Line
     Substitution – Variables all disappear – Left with True Sentence
     Elimination – Variables all disappear – Left with True Sentence

       No solutions = Inconsistent
    Graphing - Parallel Lines
    Substitution – Variables all disappear – Left with False
Sentence
    Elimination – Variables all disappear – Left with False Sentence
EXAMPLE 1        A linear system with no solution

Show that the linear system has no solution.

3x + 2y = 10                       Equation 1

3x + 2y = 2                        Equation 2

SOLUTION
METHOD 1        Graphing

Graph the linear system.
EXAMPLE 1      A linear system with no solution

The lines are parallel because they have the same
slope but different y-intercepts. Parallel lines do not
intersect, so the system has no solution.
EXAMPLE 1     A linear system with no solution

METHOD 2     Elimination

Subtract the equation.
3x + 2y = 10
3x + 2y = 2
0 = 8           This is a false statement.

The variables are eliminated and you are left with a
false statement regardless of the values of x and y.
This tells you that the system has no solution.
EXAMPLE 2      A linear system with infinitely many solutions

Show that the linear system has infinitely many
solutions.

x – 2y = – 4              Equation 1

y = 1x + 2               Equation 2
2

SOLUTION

METHOD 1 Graphing

Graph the linear system.
EXAMPLE 2     A linear system with infinitely many solutions

The equations represent the same line, so any point on
the line is a solution. So, the linear system has
infinitely many solutions.
EXAMPLE 2       A linear system with infinitely many solutions

METHOD 2      Substitution

Substitute 1 x + 2 for y in Equation 1 and solve for x.
2
x – 2y = – 4            Write Equation 1

x – 2 1x + 2 = – 4             Substitute 1 x + 2 for y.
2                                   2

–4= –4                   Simplify.

The variables are eliminated and you are left with a
true statement regardless of the values of x and y.
This tells you that the system has infinitely many
solutions.
GUIDED PRACTICE       for Examples 1 and 2

Tell whether the linear system has no solution or
infinitely many solutions. Explain.

1.     5x + 3y = 6            Equation 1
– 5x – 3y = 3            Equation 2

METHOD 2 Elimination
Subtract the equations.
5x + 3y = 6
– 5x – 3y = 3
0=9              This is a false statement.
GUIDED PRACTICE       for Examples 1 and 2

The variables are eliminated and you are left with a
false statement regardless of the values of x and y.
This tells you that the system has no solution.
GUIDED PRACTICE             for Examples 1 and 2

2. y = 2x – 4                      Equation 1

– 6x + 3y = – 12                 Equation 2

METHOD 2        Substitution

Substitute 2x – 4 for y in Equation 2 and solve for x.

– 6x + 3y = – 12          Write Equation 2

– 6x + 3(2x – 4) = – 12          Substitute (2x – 4) for y.

– 12 = – 12         Simplify.
GUIDED PRACTICE       for Examples 1 and 2

The variables are eliminated and you are left with a
true statement regardless of the values of x and y. This
tells you that the system has infinitely many solution.
EXAMPLE 3          Identify the number of solutions.

Without solving the linear system, tell whether the
linear system has one solution, no solution, or
infinitely many solutions.
a. 5x + y = – 2                  Equation 1
–10x – 2y = 4                  Equation 2

SOLUTION
Write Equation 1 in slope-
y = – 5x – 2                 intercept form.
Write Equation 2 in slope-
y = – 5x – 2
intercept form.
Because the lines have the same slope and the same
y-intercepts,the system has infinitely many solutions.
They are the same line.
EXAMPLE 3       Identify the number of solutions.

b. 6x + 2 y = 3                Equation 1
6x + 2 y = – 5              Equation 2

SOLUTION

b. y = – 3x + 3        Write Equation 1 in slope- intercept form.
2
y = – 3x – 5       Write Equation 2 in slope-intercept form.
2

Because the lines have the same slope but different y-
intercepts, the system has no solution. They are parallel
lines.
EXAMPLE 4      Write and solve a system of linear equations

ART

An artist wants to sell prints   Regular    Glossy    Cost
of her paintings. She orders
45        30      \$465
a set of prints for each of
two of her paintings. Each          15        10      \$155
set contains regular prints
and glossy prints, as shown
in the table. Find the cost of
one glossy print.
EXAMPLE 4      Write and solve a system of linear equations

SOLUTION

STEP 1

Write a linear system. Let x be the cost (in dollars) of
a regular print, and let y be the cost (in dollars) of a
glossy print.

45x + 30y = 465         Cost of prints for one painting

15x + 10y = 155         Cost of prints for other painting
EXAMPLE 4      Write and solve a system of linear equations

STEP 2

Solve the linear system using elimination.

45x + 30y = 465                    45x + 30y = 465

15x + 10y = 155                   – 45x – 30y = – 465
0=0

There are infinitely many solutions, so you cannot
determine the cost of one glossy print. You need more
information.
GUIDED PRACTICE         for Examples 3 and 4

3. Without solving the linear system, tell whether it
has one solution, no solution, or infinitely many
solutions. one solution.
x – 3y = – 15                    Equation 1
2x – 3y = – 18                    Equation 2

SOLUTION
x+5                 Write Equation 1 in slope-
y=                        intercept form.
3
2x + 18              Write Equation 2 in slope-
y=                        intercept form.
3