Transcript Substitution and Elimination

Section 10.1
Systems of Linear Equations;
Substitution and Elimination
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A movie theater sells tickets for $9.00 each, with seniors receiving a
discount of $2.00. One evening the theater took in $4760 in revenue.
If x represents the number of tickets sold at $9.00 and y the number
of tickets sold at the discounted price of $7.00, write an equation that
relates these variables.
Suppose that we also know that 600 tickets were sold that evening.
Can you write another equation relating the variable x and y?
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Does a Solution Exist?
• If a system has at least one solution
– System is consistent
• If a system has no solution,
– It is called inconsistent
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Solving System Graphically
• Solve each equation for y
– Enter the first equation as y1
– Enter the second equation as y2
– Find the intersection of y1 and y2
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2 x  y  13
Solve: 
4 x  9 y  7
Y1 =
Y2 =
Solution = (
,
)
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Verifying your answer…
• Substitute your answer into the original
equations and see if you get the correct
number.
2 x  y  13
Solve: 
4 x  9 y  7
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Your turn
• Is (3,3,0) a solution to
x  y  z  6

3 x  2 y  4 z  9
x  y  z  0

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Homework
• Page 748
– 1,4, 6,
– Solve graphically by finding the Intersection
• 9,10,16,18,26,28,30,
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Remember…
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OBJECTIVE 1
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Procedure
• Solve one of the equations for either x or y … it doesn’t matter
which equation or which variable you solve for.
– Select the variable with a coefficient of 1
• Substitute that solved equation into the other equation to
find the other variable
• Then take that value and put it into the other original
equation.
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2 x  y  13
Solve: 
4 x  9 y  7
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Your turn
• Solve the system by Substitution Method
2 x  y  3

3x  2 y  13
(-1,5)
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OBJECTIVE 2
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Procedure
Be careful of “sign”
errors !!!
• Multiply (or divide) each side of one of the equations by the
same number (not 0)
– It does not matter which equation or what you multiply by
– The idea is to make one of the variables in each of the
equations have the same coefficient
– If possible, try to eliminate the set of variables with
opposite signs.
• Add (or subtract) the two equations so that one of the
variables is eliminated
• Solve for the variable that is left
• Substitute that value into one of the original equations.
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Method of Elimination
• Solve
2 x  3 y  1

  x  y  3
(2,-1)
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y
3
 x   1
Solve:  2
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16 x  3 y  28
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Your turn #2
• Solve the system by elimination
x y
 2  3  1

 x  y   17
 3 2
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(4,-3)
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A movie theater sells tickets for $9.00 each, with seniors receiving a
discount of $2.00. One evening the theater sold 600 tickets and took
in $4760 in revenue. How many of each type of ticket were sold?
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