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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3

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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5

6

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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 13Solve:

4 9 7

x y

x y

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 13Solve:

4 9 7

x y

x y

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Your turn

• Is (3,3,0) a solution to

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3 2 4 9

0

x y z

x y z

x y z

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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 13Solve:

4 9 7

x y

x y

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Your turn

• Solve the system by Substitution Method

(-1,5)

2 3

3 2 13

x y

x y

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OBJECTIVE 2

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Procedure• 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.

Be careful of “sign” errors !!!

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Method of Elimination

• Solve

(2,-1)

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2 3 1

3

x y

x y

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Solve: 2 816 3 28

yx

x y

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Your turn #2

• Solve the system by elimination

(4,-3)

12 3

17

3 2 6

x y

x y

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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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