Transcript Alg2-Ch3-Sect1_2-Power_Point_Lesson

3-1
3-1,2
Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Warm Up (Slide #2-3)
Objective and Standards (Slide #4)
Vocab (Slide #5–8)
Lesson Presentation (Slide #9–31)
Text Questions (NONE)
Worksheets 3.1A, 3.2A (Slide #32)
Lesson Quiz (Slide #33-34)
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Algebra
Holt
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3-1
3-1,2
Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Warm Up
Use substitution to determine if (1, –2) is an
element of the solution set of the linear
equation.
1. y = 2x + 1 no
2. y = 3x – 5 yes
Write each equation in slope-intercept form.
4. 4y – 3x = 8
3. 2y + 8x = 6
y = –4x + 3
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3-1
3-1,2
Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Warm Up
Determine if the given ordered pair is an
element of the solution set of
2x – y = 5
3y + x = 6
1. (3, 1) yes
2. (–1, 1) no
Solve each equation for y.
3. x + 3y = 2x + 4y – 4 y = –x + 4
4. 6x + 5 + y = 3y + 2x – 1 y = 2x + 3
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3-1
3-1,2
Using Graphs and Tables
Solving
Linear
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SystemsSystems
Objectives
1. Solve systems of linear equations with:
•Graphs and tables
•Substitution
•Elimination
2. Determine whether there will be one, none, or an
infinite number of solutions by noting characteristics of
each equation.
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3-1,2
Using Graphs and Tables
Solving
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to
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SystemsSystems
Vocabulary
system of equations
linear system
substitution
elimination
linear combinations
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Using Graphs and Tables
Solving
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A system of equations is a set of two or more
equations containing two or more variables. A linear
system is a system of equations containing only linear
equations.
A line is an infinite set of points that are solutions to a
linear equation. The solution of a system of equations is
the set of all points that satisfy each equation.
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Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
There are two aspects of substitution:
In one, a possible solution (ordered pair) is given and you
simply substitute its x and y values into each equation to
see if that point satisfies both.
In the other, you substitute the equivalent expression for
a variable from one equation into the other equation,
solve for one variable, then use that value to solve for the
other variable.
(I know…it sounds all so confusing, but it’s really easy.)
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Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
You can also solve systems of equations with the
elimination method.
With elimination, you get rid of one of the variables by
adding or subtracting equations.
You may have to multiply one or both equations by a
number to create variable terms that can be eliminated.
Reading Math
The elimination method is sometimes called the
addition method or linear combinations.
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3-1,2
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Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Points to remember about linear equations and
systems:
On the graph of the system of two equations, the
solution is the set of points where the lines
intersect.
A point is a solution to a system of equation if the
x- and y-values of the point satisfy both equations.
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3-1,2
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Using Graphs and Tables
Solving
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to
Solve Linear
SystemsSystems
Example 1A: Verifying Solutions of Linear Systems
To see if a given point is a solution to a linear system,
substitute the (x,y) values into both equations.
For example… “Is (1,3) the solution to this linear system?”
(1, 3);
x – 3y = –8
3x + 2y = 9
Ans: YES
Holt Algebra 2
3-1,2
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Using Graphs and Tables
Solving
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Solve Linear
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Check It Out! Example 1b
Is (5,3) an element of the solution set for the
system of equations?
(5, 3);
6x – 7y = 1
3x + 7y = 5
Ans: NO
Holt Algebra 2
3-1,2
3-1
Using Graphs and Tables
Solving
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Solve Linear
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Example 2A: Solving Linear Systems by Using Graphs
and Tables on your graphing calculator
Solve the system. Check your answer.
First, solve each equation for y, then graph both:
2x – 3y = 3
y+2=x
On the graph, the lines
appear to intersect at
the ordered pair (3, 1)
Use the calculator’s
“Table” or “Trace”
function to verify.
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Sol’n: (3,1)
3-1,2
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Using Graphs and Tables
Solving
Linear
to
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SystemsSystems
Check It Out! Example 2b
Use a graph and a table to solve the system.
Check your answer. First, solve each equation for y.
x+y=8
2x – y = 4
Sol’n: (4, 4)
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Different Slopes
will have
ONE solution
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Same Slopes
and Same Y-int.
will have
INFINITE Sol’ns.
Same Slopes
but Different Y-int.
will have
NO Solutions.
3-1,2
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Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Remember!
An identity, such as 0 = 0, 8 = 8, -7 = -7, etc…
is always true and indicates infinite solutions.
A contradiction, such as 1 = 3, 5 = 9, -8 = 8, etc…
is never true and indicates no solution.
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3-1,2
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Using Graphs and Tables
Solving
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Solve Linear
SystemsSystems
Example 4: Summer Sports Application
One golf course charges $20 to rent golf clubs plus
$55 per hour for golf cart rental.
A different course charges $35 to rent clubs plus
$45 per hour to rent a cart.
Both places allow rentals in ½hr. increments.
Q: For what number of hours is the cost of renting
clubs and a cart the same for each course?
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Using Graphs and Tables
Solving
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Example 4 Continued
Step 1 Write an equation for the cost of renting clubs
and a cart at each golf course.
Let x represent the number of hours and y represent
the total cost in dollars.
City Park Golf Course: y = 55x + 20
Sea Vista Golf Course: y = 45x + 35
Sol’n: 1.5hrs
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3-1,2
3-1
Using Graphs and Tables
Solving
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Check It Out! Example 4
Ravi is comparing the costs of long distance
calling cards. To use card A, it costs $0.50 to
connect and then $0.05 per minute. To use
card B, it costs $0.20 to connect and then
$0.08 per minute. For what number of minutes
does it cost the same amount to use each card
for a single call?
Step 1 Write an equation for the cost for each of the
different long distance calling cards.
Let x represent the number of minutes and y represent
the total cost in dollars.
Card A: y = 0.05x + 0.50
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Card B: y = 0.08x + 0.20
3-1,2
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Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Check It Out! Example 4
Card A: y = 0.05x + 0.50
Card B: y = 0.08x + 0.20
Step 2 Here’s a situation where, since both are equal to
“y”, you can set the two equations equal to each other.
.05x + .50 = .08x + .20
-.05x
-.05x
-.20
-.20
.30 = .03x
÷.03 ÷.03
10 = x
So, both plans are the same for a 10minute call
Holt Algebra 2
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Example 1A: Solving Linear Systems by Substitution
Use variable substitution to solve the system:
y= x–1
x+y=7
Step 1: Substitute the equivalent expression for “y”
from the first equation in place of “y” in the second
equation and solve for “x”.
x+y=7
x + (x – 1) = 7
2x – 1 = 7
2x = 8
x=4
Then…
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Solving Linear Systems
Using Algebraic Methods
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3-2 to Solve Linear Systems
Example 1A Continued
Step 2: Substitute the x-value into one of
the original equations to solve for y.
y=x–1
y = (4) – 1
y=3
The solution is the ordered pair (4, 3).
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Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Example 1A Continued
Check A graph or table supports your answer.
Holt Algebra 2
Solving Linear Systems
Using Algebraic Methods
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3-2 to Solve Linear Systems
Example 1B: Solving Linear Systems by Substitution
Use substitution to solve the systems of equations.
2y + x = 4
y = 2x – 1
5x + 6y = –9
3x – 4y = 7
3x + 2y = 26
Sol’n: (3, 1/2)
Sol’n: (4,7)
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2x – 2 = –y
Sol’n: (3,-4)
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Check It Out! Example 1a Continued
Check A graph or table supports your answer.
Holt Algebra 2
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Example 2A: Solving Linear Systems by Elimination
3x + 2y = 4
3x + 5y = –16
4x – 2y = –18
2x + 3y = –9
Sol’n: (–2, 5)
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Sol’n: (3,-5)
4x + 7y = –25
–12x –7y = 19
Sol’n: (.75, -4)
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Check It Out! Example 2b
Use elimination to solve the system of equations.
5x – 3y = 42
8x + 5y = 28
Sol’n: (6,-4)
Holt Algebra 2
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Example 3: Solving Systems with Infinitely Many or
No Solutions
3x + y = 1
56x + 8y = –32
6x + 3y = –12
2y + 6x = –18
Sol’n: NONE
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7x + y = –4
Sol’n: INFINITE
2x + y = –6
Sol’n: NONE
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Example 4: Zoology Application
A veterinarian needs 60 pounds of dog food
that is 15% protein. He will combine a beef
mix that is 18% protein with a bacon mix that
is 9% protein. How many pounds of each does
he need to make the 15% protein mixture?
Let x present the amount of beef mix in the mixture.
Let y present the amount of bacon mix in the mixture.
Write one equation based on the amount of dog food
Write another equation based on the amount of protein
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THEN…
Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Example 4 Continued
Solve the system.
x + y = 60
0.18x +0.09y = 9
Sol’n: (40, 20)
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Solving Linear Systems
Using Algebraic Methods
3-1,2
3-2 to Solve Linear Systems
Check It Out! Example 4
A coffee blend contains Sumatra beans which
cost $5/lb, and Kona beans, which cost
$13/lb. If the blend costs $10/lb, how much
of each type of coffee is in 50 lb of the blend?
Let x represent the amount of the Sumatra beans in the blend.
Let y represent the amount of the Kona beans in the blend.
Write one equation based on the amount of each bean
Write another equation based on cost of the beans:
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THEN…
Solving Linear Systems
Using Algebraic Methods
3-2 to Solve Linear Systems
3-1,2
Check It Out! Example 4 Continued
Solve the system.
x + y = 50
5x + 13y = 500
Sol’n: (18.75, 31.25)
Holt Algebra 2
Solving Linear Systems
Using Algebraic Methods
3-2 to Solve Linear Systems
3-1,2
Holt Algebra 2
3-1,2
3-1
Using Graphs and Tables
Solving
Linear
to
Solve Linear
SystemsSystems
Lesson Quiz: Part I
Use substitution to determine if the given
ordered pair is an element of the solution set of
the system of equations.
x + 3y = –9
x+y=2
2. (–3, –2)
1. (4, –2)
y – 2x = 4
y + 2x = 5
Solve the system using a table and graph.
Check your answer.
x+y=1
3.
3x –2y = 8
Holt Algebra 2
3-1,2
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Using Graphs and Tables
Solving
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Solve Linear
SystemsSystems
Lesson Quiz: Part II
Which system has NO solution and which has
INFINITE solutions.
y + 2x = –10
–4x = 2y – 10
5.
4.
y + 2x = –10
y + 2x = –10
6. Kayak Kottage charges $26 to rent a kayak plus
$24 per hour for lessons. Power Paddles charges
$12 for rental plus $32 per hour for lessons. Both
places allow rentals in 15min (1/4hr) For what
number of hours is the cost of equipment and
lessons the same for each company?
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