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P1258790476YwjgX

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... of 537 electoral votes, the other equation for the system of linear equations is: ... Created by Bridget Feneis, Ocean County College. Okay, you're on your own... – PowerPoint PPT presentation

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Title: P1258790476YwjgX


1
Chapter 5 Systems of Linear Equations
2
  • Determine if the following ordered pair (point)
    is a solution to the
  • following system of equations

Solution
How do you know if a point is a solution to these
two equations?
Plug the point into both equations to see if the
point makes both equations true. The following
will show this
Yes, 7 equals 7, so the point is a solution to
the 1st equation. Now lets try the 2nd equation.
3
Yes, 3 equals 3, so the point is also a solution
to the 2nd equation.
Since the point was a solution to BOTH equations,
the point is a solution to the system of
equations.
4
2. Determine if the following ordered pair
(point) is a solution to the following system
of equations
Solution
No, -10 does not equal -2, so the point is not a
solution to the 2nd equation.
Since the point was not a solution to BOTH
equations, the point is not a solution to the
system of equations.
Yes, -8 equals -8, so the point is a solution to
the 1st equation. Now lets try the 2nd equation.
5
3. Solve the following system of equations
graphically
Solution
We will graph each equation using the intercept
method from Chapter 4.
Again, the intercept method finds the
x-intercept (x,0), the y-intercept (0,y), and
a third point called the check point.
Lets graph the 1st equation
Lets plot these and draw the line
6
Again, well use the intercept method to graph
the 2nd equation
The solution to the system is (4,2)! This point
is common to both equations. That is, it is a
solution to both equations. Note This is the
point of intersection.
Again, lets plot these and draw the line (in
purple) above
7
4. Solve the following system of equations
graphically
Solution
Again, we will graph each equation using the
intercept method. Except this time, I will find
the x- and y-intercept but will not find the
check point.
Lets graph the 1st equation
And this line looks like
8
Again, well use the intercept method to graph
the 2nd equation
Uh-oh, there is no point that is common to both
lines these lines are parallel. The solution
is no solution. Note When the system has no
solution, the system is called inconsistent.
Again, lets plot these and draw the line (in
purple) above
9
5. Solve the following system of equations
graphically
Solution
Again, we will graph each equation using the
intercept method. Again, I will find the x- and
y-intercept but will not find the check point.
Lets graph the 1st equation
And this line looks like
10
Again, well use the intercept method to graph
the 2nd equation
Oh goodness, they are the same line. Thus, the
solution is all of the points on the line
(infinitely many ordered pair solutions). Note
When the system has all points on the line as a
solution, the system is called dependent.
Again, lets plot these and draw the line (in
purple) above
11
What are the advantages and disadvantages to
solving a system using the graphing method?
Advantage When you graph, you can SEE what is
happening with the linesare they intersecting?
are they parallel? are they the same line?
  • Disadvantage
  • If the solution is a fraction, it will be
    difficult to write the answer from the graph.
    For example, if the solution is , how
    would you know this?
  • What happens if the point of intersection
    happens at (1000,2000)- Is your graph that big?
  • What happens if the lines look parallel but
    they really are not.

So, you can see some problems with the graphing
method
12
So heres the Substitution Method - an algebraic
method
6. Solve the following system of linear equations
by the Substitution Method
Solution
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7. Solve the following system of linear equations
by the Substitution Method
Solution
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8. Solve the following system of linear equations
by the Substitution Method
Solution
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21
  • Solve the following system of linear equations by
    the
  • Elimination Method

Solution
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10. Solve the following system of linear
equations by the Elimination Method
Solution
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11. Solve the following system of linear
equations by the Elimination Method
Solution
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Summary of Solving a System of Linear Equations
  • You get a solution, (x,y). Graphically, this
    means the lines intersect at this point (x,y).
    This system is called consistent.
  • The variable terms cancel and you are left with a
    false statement, such as 07. This means the
    system has NO SOLUTION. Graphically, this means
    the lines are parallel. Additionally, the system
    is called inconsistent.
  • The variable terms cancel and you are left with a
    true statement, such as 00. This means the
    system has INFINITELY MANY SOLUTIONS (the points
    on the line). Graphically, this means the lines
    are the same. Additionally, the system is called
    dependent.

Now for the Application Problems!
28
Solution
29
When the ferry travels downstream, the speed of
the ferry in still water is increased by the
speed of the current, hence the rate
When the ferry travels upstream, the speed of the
ferry in still water is decreased by the speed of
the current, hence the rate
30
Direction Rate Time Distance
Downstream 4
Upstream 6
31
Distance (from table) Distance (from problem)
Using the elimination method, multiply (a) by 3
and (b) by 2
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33
13. In the 2000 presidential election, George W.
Bush received 5 more electoral votes than Al
Gore. Together, the two presidential candidates
received a total of 537 electoral votes. How
many electoral votes did each receive? (Source
World Almanac and Book of Facts, 2002.)
Solution
Since George Bush received 5 more electoral votes
than Al Gore, one equation for the system of
linear equations is
Since the two candidates received a total of 537
electoral votes, the other equation for the
system of linear equations is
34
Thus, the system of linear equations to solve is
Since equation (a) is already solved for b, well
use substitution.
And since George W. Bush received 5 more
electoral votes, George W. Bush received
2665271 electoral votes.
Thus, Al Gore received 266 electoral votes.
35
14. In the Bay Head candy shop, double-dip mint
chocolates cost 10.00 per pound and swedish
fish cost 6.00 per pound. Mary, who loves
both, notices a mixture of the two selling for
7.00 per pound. She purchases 4 pounds and then
wonders how many pounds of each she received.
Solution
Since 4 pounds of the mixture were purchased, one
equation for the system of linear equations is
Now for the value (or cost) of each candy. If
the mints cost 10.00 per pound and x represents
the number of pounds you have, then the total
value (or cost) of the mints is
36
Using this same thinking, if the swedish fish
cost 6.00 per pound and y represents the number
of pounds you have, then the total value (or
cost) of the swedish fish is
Since the mixture was purchased at 7.00 per
pound and 4 pounds were purchased, the total cost
was (7.00)(4) 28.00 Thus the total value
equation leads to the other equation for the
system of linear equations
Thus, the system of linear equations to solve is
37
Substitution or elimination may be used to solve
this system. I chose elimination.
Note In the book, this problem was approached
using a table to set up the information.
Thus, 3 pounds of swedish fish were purchased.
And 1 pound of double-dip mint chocolates.
38
Okay, youre on your own You know what you need
to do!
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